The best place to look for your McAndrew's would be northeastern County Mayo. In Irish, the surname McAndrew is Mac Aindriu. It was adopted by a branch of the Barrett family in Mayo. The Barrett’s were an Anglo-Norman family which became fully Hibernicized (the name was sometimes re-Anglicized as FitzAndrew). The McAndrew’s became like a native Irish sept (or “sub-tribe”), having a well-defined territory on the eastern side of Lough Conn in northeastern Mayo (in the area of the current parishes of Backs, Attymass, and other nearby parishes). In Griffiths Valuation, in the mid-1800's, there were 232 McAndrew tenants or landholders listed in all of Ireland, of whom 186 were in County Mayo, and 19 in adjoining parts of County Sligo, with a few also in Roscommon, and all 16 of the McAndrew births registered in 1890 in Ireland were in Mayo.
The Catholic parish records for Attymass and Backs don't go far enough back for you to find baptismal records for your Richard or Anthony, but there were/are also a lot of McAndrew's in the parish of Kilmoremoy (Ballina), and the records there go back to 1823, so you might be able to find younger siblings of Richard and Anthony there, or maybe even Richard or Anthony themselves, if they emigrated in their late teens.. Those records are available at this link:
You'll see an interactive map there which will allow you to reach the info for the other nearby parishes.
As it happens, one of my great-great-great grandfathers was namd Thomas McAndrew, and he was born circa 1780, so about the same time as your Thomas would likely have been born. I know of two of his children, my great-great-grandmother, Bridget, and her brother, John, but it's likely that there were other children. They lived in the townland of Coolcastle (Coolcashla), in the parish of Killasser, which you'll see from that map adjoins Attymass to the southeast. There were also McAndrew families in several neighboring townlands, but I'm not able to establish most of the connections, because the records don't go back far enough to do that.