The surname could certainly be found in Carrickfergus in the 1800s, however with the limited knowledge you have of him, it could prove difficult to link anysuch persons directly to your chap.
For example, the Pre-1858 Wills and Admons index at PRONI has a Fanny Mayne, Carrickfergus, 1831 and a Samuel Mayne, Carrickfergus, 1835 listed, albeit the original documents do not exist. PRONI also have the will indexed for a James Mayne of Carrickfergus, who died in 1869 and was married to a Jenny. The image of same does not mention children.
Newspapers have mention of an 1868 death in Carrickfergus of an Andrew Mayne, who was born in 1803. He was apparently late of H.M. Customs, Liverpool. Images are not yet available online for the death registration, so no further details available.
The Griffiths Valuation of mid 1800s also has examples of the surname in Carrickfergus, eg a John Mayne in the town of Carrickfergus in 1861.
Another example of the surname in the area albeit later, at 1901, can be seen at:
https://www.census.nationalarchives.ie/pages/1901/Antrim/Carrickfergus/North_Street/993064/ .
Have you or any other family descended from Richard, considered going down the DNA route to perhaps assist in linking with others of the Mayne surname?