Shipping Intelligence. Arrivals, The Sydney Morning Herald, 10 March 1843, p 2, http://nla.gov.au/nla.news-article12411481:

From Liverpool, same day, via the Cape of Good Hope, having left the former the 2nd November, the latter the 17th January, the ship Templar, 565 tons, Captain Brown, with a general cargo. Passengers—Most Rev. Dr. Polding, Rev. Dr. Gregory, Rev. Messrs. [...] McCarthy, [...]

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Shipping Intelligence. Arrivals, The Colonial Observer, Sydney, 11 March 1843, p 877, http://nla.gov.au/nla.news-article226359488:

9. The ship Templar, Captain Brown, from Liverpool, 2nd November, and Cape of Good Hope, with merchandise. Passengers—Bishop Polding, Roman Catholic priests, viz. : Messrs. Gregory, [...] Macarthy, [...]

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Shipping Intelligence. Arrivals, The Sydney Morning Herald, 11 September 1843, p 2, http://nla.gov.au/nla.news-article12415730:

From Norfolk Island, same day, the brig Governor Phillip, 198 tons, Captain Boyle. Passengers,—Rev. Mr. McCarthy, [...]

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Domestic Intelligence. The Roman Catholics and the Church Act, The Sydney Morning Herald, 2 January 1844, p 2, http://nla.gov.au/nla.news-article28651191:

A public meeting was held yesterday afternoon, pursuant to announcement, at the Roman Catholic Church of St. Mary, for the purpose of taking into consideration the proposed expenditure of the £30,000 devoted to ecclesiastical purposes. [...]

The Rev. Mr. McCarthy after a few preliminary remarks moved the second resolution, which was to the following effect:—
That we do record our determination to abide by, and to uphold the Church Act, which experience has abundantly proved to be admirably adapted to the circumstances of this colony, and to resist all restraints and encroachments on its provisions by all constitutional means. [...]

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Meeting of Catholics, Morning Chronicle, Sydney, 3 January 1844, pp 2-3, http://nla.gov.au/nla.news-article31741853:

"Pursuant to notice, a meeting of the Catholic community was held on New Year's day, at St. Mary's Cathedral, for the purpose of taking into consideration the expenditure of £30,000, which was appropriated for ecclesiastical purposes.

His Grace the Most Rev. Dr. Polding, who was attended by the Very Rev. the Vicar General, Rev. John McEncroe, Rev. Dr. Gregory, Rev. Mr. Bourgeois, Rev. Mr. Young, Rev. Mr. McCarthy and Rev. Mr. Coffey, took the chair a little after one o'clock.

[...] The Rev. Mr. McCarthy, in proposing the second resolution, said that it spoke abundantly for itself, and that after the full and satisfactory explanation given by his Grace the Archbishop of the objects of the meeting, little need be said by him. He thought the Catholic clergy and laity ought to unite together to uphold their Archbishop in his exertions to maintain their religion in its perfection, and not allow of any infringement upon those rights to which they were entitled. The Rev. gentlemen concluded by reading the following resolution:—
"That we do record our determination to abide by, and to uphold the Church Act, which experience has abundantly proved to be admirably adapted to the circumstances of this colony, and to resist all restraints and encroachments on its provisions by all constitutional means. [...]

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NSW death registration: Thomas F. McCarthy, age 42, no. 334/1844 V1844334 146.

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Death, Morning Chronicle, Sydney, 3 July 1844, p 2, http://nla.gov.au/nla.news-article31743107:
We have to record with deep regret the death of the Rev. Thomas Macarthy. This estimable ecclesiastic was professed at an early period in the venerable order of the Capuchins, of the institute of St. Francis. Having completed his course of theological studies in Rome, he returned to Ireland: and in Cork, Kilkenny, and Dublin, laboured most zealously and effectively, to bring souls to the practice of a holy life. He was the beloved friend and disciple of the illustrious Father Mathew, whose consent to his departure for these shores was at first reluctantly given. The ardent desire, however, which the Rev. Mr. Macarthy expressed to devote himself without reserve to the salvation of souls, far from friends -- from home and country -- obtained not his consent only, but his warmest admiration. The lamented deceased accompanied his Grace the Archbishop on his return to Sydney. During the voyage his amiable gentility, his piety and virtue, endeared him to all, but especially to his ecclesiastical superior -- by whom his loss is most severely felt. He returned some months since from Norfolk Island, in a delicate state of health, and was placed at Windsor. The most sanguine hopes were entertained of his perfect recovery; these, alas! were too soon dissipated by the symptoms of fatal decline. The severity of long protracted sickness was endured with the greatest patience, and in the presence of his dear friend and confessor, Mr. Coffey, and the Sisters of Charity, retaining his senses and self-possession to the last, he calmly resigned his pure soul into the hands of his Creator about 1 p.m. on Wednesday, the 26th ultimo. A solemn Dirge and Mass were celebrated on Friday, the 28th, at which the Rev. clergy of Sydney, Liverpool, Windsor, and Campbelltown assisted. The Archbishop presided at the office, and performed the funeral service. Mass was celebrated by the Rev. Mr. Coffey. The procession to the grave was conducted with the greatest decorum. The large crowd attending seemed to be duly impressed with the solemnity of the occasion which had brought them together; and it may be hoped that few retired without some desire of preparing by a virtuous life, for that happiness which it is our comfort to believe a merciful God has bestowed on that zealous minister, who left all things for his sake. The Rev. Thomas Macarthy was in the 42nd year of his age.

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Original Poetry. Lines to the Memory of the Rev. Thomas McCarthy, Who Died at Parramatta on the 26th June, 1844, Morning Chronicle, Sydney, 20 July 1844, p 5, http://nla.gov.au/nla.news-article31743198:

"Fidelis servus et prudens quem constituit Dominus super familiam suam." [...]
Philos.
July 15, 1844.

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Patrick Francis Cardinal Moran, History of the Catholic Church in Australasia, 1895, p 267:
Things were in this unsettled state when the Bishop [Willson], accompanied by Father Hotham, proceeded to Sydney in the beginning of August, 1844, to assist at the Episcopal consecration of the newly appointed Bishop of Adelaide, and also to take part in the Provincial Synod, which had been summoned by the Archbishop. He arrived in Sydney on the 16th of August. Four days later he accompanied the Archbishop to Parramatta, where the foundation stone was blessed of a mortuary chapel , the first erected in the Australian Church, and intended as a memorial of the Rev. Thomas McCarthy, who had died in Parramatta on the 26th of the preceding June.

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A plaque was installed in the Parramatta mortuary chapel in August 1994 to mark his grave (date of death stated as 26 July, rather than the actual date, 26 June 1844).