Some Irish names beginning with H did not start out that way. For Haffy, Hallinan, Hallisey, Hanlon, Hanly, Hannigan, Hanrahan, Hartigan, please check under A.
For Hegarty, check under E.
For Heffernan, check under I.
For Hogan, Horan, and Honora, check under O.
For Houlihan, check under U.
For K and Q names, check under C.
For J names, check under I.
For W names, check under U.
There are likely no names from Irish that begin with V, X, Y or Z. There aren't any here!