Translation according to P. W. Joyce:
Cloonavihony in Galway; badly anglicised, as many good authorities spell it Cloonacavohony, pointing to the true Irish form, Cluain-Mhic-Mhathghamhna, MacMahon's meadow. See Mac [reproduced below].
Mac, a son. As part of a family name, it very often enters into place-names. It suffers many changes, chiefly consisting of clipping off some of its letters; so that it appears as ma, mic, ac or ack, ic or ick, c or k. All those appear in scores of the names in this book. See vol. ii. p. 143. In all these variations the c is often made g - mag: an allowable change.