"O Master Lydgate! the sweetest source of famous rhetoric, and of ballad royal the chief originator."
Rhyme royal was a standard form of English narrative verse from Chaucer down to the fifteenth century: seven lines, usually of five but occasionally of four iambic feet, rhyming a-b-a-b-b-cc.
Ballad royal was an English variant of the French 'fixed form' ballade: four stanzas of rhyme royal, with all four stanzas using the same rhymes and ending in the same refrain.
However, nomenclature was flexible in the days of Lydgate and his successors; Hawes is probably speaking of Lydgate's huge output of rhyme royal rather than of works in the short form. The stage directions to Lydgate's Bycorne and Chychevache employ the term balade for what we would call stanza, e.g. First there shal stonde an ymage in poete-wyse seying thees thre balades, which is followed by three stanzas of rhyme royal.