'The Chaplet', South Australian Register, Adelaide, 3 May 1871, p 5, http://nla.gov.au/nla.news-article39241929:

This Roman Catholic monthly has been enlarged to the size of its predecessor the Southern Cross, and a copy of the issue, dated April 29, has been received. From it we glean the following items :— Bishop Sheil, who on account of delicate health left for Melbourne on April 11, is expected back within a few days. [...] During the month several changes have taken place in the clerical staff of the diocese. The Rev. J. Quinlan is now stationed in Willunga, and the Rev. M. Kennedy in Adelaide. The Rev. P. Byrne goes to Mount Barker with the Rev. J. Maher, and the Rev. P. Corcoran to Kadina in place of Father Quinlan.

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Clerical Changes, The Irish Harp and Farmers' Herald, Adelaide, 1 June 1872, p 5, http://nla.gov.au/nla.news-article224364273:

[...] The Rev. P. Corcoran from Kadina to St. Patrick's, West Terrace. [...]

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Consecration of the Roman Catholic Bishop, South Australian Register, Adelaide, 6 November 1873, p 5, http://nla.gov.au/nla.news-article39301141:

On Sunday, November 2, many South Australians had an opportunity of witnessing one of the most impressive ceremonials of the Roman Catholic Church, an event which may be seen perhaps but once in a lifetime, and which certainly will not easily be forgotten by those who were spectators on the occasion. The advancement of a priest to the dignity of the Episcopate must always be regarded as a circumstance of importance, and this being the first time in the history of the diocese of Adelaide that the chief pastor of the Romish Church here has been consecrated amongst his flock, gave additional interest to the event. This was intensified by the presence of the Venerable Roman Catholic Primate of Australia and three prelates from the sister colonies, who formed such an assemblage of the pillars of the Church as has never before been seen in this province. The solemn act of consecration took place at St. Francis Xavier's Cathedral, before a crowded congregation, including many members of other denominations. [...] At 10 o'clock a procession entered. [...] Next appeared the assistant consecrators, Dr. Murphy, Bishop of Hobart Town, with his chaplain, the Rev. P. Kennedy ; Dr. Quinn, Bishop of Bathurst, and his chaplain, the Rev. P. Corcoran. The prelates were attired in mitres and magnificent copes of cloth of gold, the upper portion embroidered with flowers in natural colours. Dr. Murray, Bishop of Maitland, was attended by the Rev. Father Bourke as chaplain, and had on the ordinary purple of a Bishop. The Archbishop (the Consecrator), who was preceded by an acolyte bearing his silver Pastoral staff, followed, attired in vestments of black satin, and supported by his chaplains, Fathers Colletti, of Sydney, and Strele, S.J. At the last came the Bishop-elect, the Very Rev. C. A. Reynolds, wearing his biretta and robed in a cope of white moire antique, worked with flowers and fringed with gold. His chaplains were the Very Rev. F. Byrne and the Rev. J. Tappeiner, S.J. [...]

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South Australia. Diocese of Adelaide. Clerical Changes, Advocate, Melbourne, 11 October 1873, p 6, http://nla.gov.au/nla.news-article170300496:

The Venerable the Archdeacon has taken charge of his old mission, St. Lawrence's, North Adelaide, and the Rev. P. Corcoran has been appointed assistant priest.

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South Australia. Diocese of Adelaide. (Harp, August 7th.). Feast of S. Ignatius, Advocate, Melbourne, 22 August 1874, p 6, http://nla.gov.au/nla.news-article170303523:

Friday last being the Feast of S. Ignatius Loyola, founder of the Society of Jesus, was observed with the customary solemnities at the Church of the Jesuit Fathers in Norwood. Masses were celebrated at the high altar and the altar of the Sacred Heart from half-past 6 to 9 o'clock; and at 10 o'clock, Pontifical Mass was celebrated by his Lordship the Bishop [...] In addition to the above, there were present [...] Revs. P. Corcoran, [...]

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South Australia. Diocese of Adelaide. (Tablet, 5th April.), Advocate, Melbourne, 13 May 1876, p 7, http://nla.gov.au/nla.news-article170431961:

We understand that the Rev. Father Corcoran, of Willunga, has resigned his mission there, and is about leaving South Australia. The rev. gentleman has been stationed in several missions of the Diocese of Adelaide during his sojourn in the colony. [...]

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Shipping Intelligence. Hobson's Bay. Arrived—May 18, The Argus, Melbourne, 19 May 1876, p 4, http://nla.gov.au/nla.news-article7439565:

Aldinga, s.s., 291 tons, Wm. M'Lean, from Adelaide 16th inst. Passengers—saloon: [...] Rev. Father Corcoran [...]

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Diocese of Ballarat, Advocate, Melbourne, 30 September 1876, p 6, http://nla.gov.au/nla.news-article170433430:

[...] The Rev. Father Corcoran officiated at Vespers, and the sermon on the occasion was preached by the Rev. Father Cleary. [...]

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Shipping Intelligence. Hobson's Bay. Cleared Out—June 20, The Argus, Melbourne, 21 June 1877, p 4, http://nla.gov.au/nla.news-article5926397:

Northumberland, s.s., 1,400 tons, H. H. Shinner, for London. Passengers— [...] Rev. P. Corcoran [...]

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Advocate, Melbourne, 23 June 1877, p 12, http://nla.gov.au/nla.news-article170436203:

The well-known favourite Blackwall liner Northumberland, under the command of Captain H.H. Shinner, again took her departure on the homeward journey on Wednesday. Among her passengers was the Rev. P. Corcoran, of the Ballarat Diocese, who goes home on a twelve months' leave of absence, and Mr. Justice Stephen and his family. Father Corcoran was presented by his Ballarat friends with a purse of sovereigns on Tuesday, and they will be delighted to welcome him back upon his return to the Diocese.

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English Shipping. Arrivals, Weekly Times, Melbourne, 13 October 1877, p 17, http://nla.gov.au/nla.news-article220464206:

Per Northumberland, s., from Melbourne.— [...] Rev. P.J. Corcoran.

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Irish death record, https://www.irishgenealogy.ie/files/civil/deaths_returns/deaths_1914/05301/4474179.pdf: Superintendent Registrar's District Waterford, Registrar's District Kilmackevoge. 1914. Deaths Registered in the District of Kilmackevoge in the Union of Waterford in the County of Kilkenny. No. 249: On 18 April 1914 at Rossinan, death of Reverend Patrick Corcoran, male, bachelor, 68 years, Roman Catholic Clergyman; of Cerebral Apoplexy, 20 days, Heart Failure, 18 hours, certified; informant: Thomas Corcoran, brother, present at death, of Rossinan, 11 June 1914.

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Irish News, The W.A. Record, Perth, 6 June 1914, p 18, http://nla.gov.au/nla.news-article212356414:

The death took place in April last, at the residence of his brother, Mr Thos. Corcoran, Rossman, Mullinavat, of the Rev. P. J. Corcoran. The deceased had been for over 30 years on the Australian mission, and came back to his native land about twelve years ago. He was brother of the Rev. Michael Corcoran, Durrow, Queen's Co., and the esteem in which he was held was evidenced by the large cortege which followed his remains to their last resting-place in the family burial ground at Mullinavat. Previous to the interment, Office and High Mass took place, the Rev. M. Corcoran being celebrant of the High Mass.

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Two Linane entries proven by above research to relate to the same person:

Fr T.J. Linane, From Abel to Zundolovich, Vol. 2: Biographies of Priests on the Australian Scene up to 1900, 1980, p 84:

Corcoran, P., Irish
An Enigma for the Columnist
This Corcoran (with the initial "P") on the list of Adelaide priests is probably identical with Patrick Corcoran of the Ballarat Diocese. If so he is a much travelled man and one who was constantly on the move and beset by poor health. He arrived in Adelaide with Bishop L.B. Sheil on 4 December 1868, and carried out assignments in Mount Gambier (1869), Adelaide Cathedral (1870), and Kadina (1871). There the trail ends ... but the timing would seem to suggest that he came across the border to Victoria. Hence ......
(Footprints, Vol. 2, No. 11)

Fr T.J. Linane, From Abel to Zundolovich, Vol. 2: Biographies of Priests on the Australian Scene up to 1900, 1980, p 84:

Corcoran, Patrick, Irish
Purchased a Church from the Wesleyans
Origin and education are unknown, and even the time of arrival. One note on the desk states he was in Port Fairy in 1872, and another record states that he arrived in Victoria about 1876. Certainly there was a Patrick Corcoran in Ballarat from 1876 to 1879, allowing for an interlude for an overseas trip, 1877-8. In 1880 he was stationed at Port Fairy, leaving there to come to the Smythesdale Parish (now Linton) where he served from 1881 to 1886. While in Smythesdale he attended to at least a dozen churches in small towns that in turn had witnessed a gold boom but which never held their populations. Probably it was at Smythesdale that the proclivity for a T.B. condition developed, as had happened with his predecessor, Fr Patrick Moore. Long horseback rides over bridle tracks — and often with damp clothes on the back — were not conducive to prolonged life. Only one achievement seems to be on record: purchasing a chapel from the Wesleyans to serve the Catholics of Skipton in their need for a church of their own. Bad health and a chest condition brought about a change to a milder climate at Inglewood, a mining town close to Bendigo. After a year's stay he again went overseas in 1887 and apparently resolved not to return to Ballarat's cold climate. Three years later he returned to New South Wales, being given a post at Camden where he remained until 1895. From there he was transferred to Liverpool, remaining or rather holding out against the T.B. condition for a brief year or so. Directories from 1898 to 1903 list the sufferer as being: on leave. A report given somewhere (which is not before me at the moment) states that he died in Ireland about 1904.
(Footprints, Vol. 2, No. 11)
Ref.: MDHC File, Torquay, per Fr. L. Gallagher P.P., Linton.