Morrissey has complete and utter control over everything to do with the lyric and vocal melody. When a song is credited as being "Morrissey/Whyte", for example, the basic music was written by Alain and arranged with the band and Moz has then written lyrics for the track.
However, Moz does have a big say on the arrangment, for example Boz has said that there were some piano parts on Crashing Bores than Moz decided didn't need to be there so off they went, and if he decides what was written to be the verse should be the middle eight then it gets changed. So you could probably fairly say that in a song written by Moz and an other, it's probably about a 65/35 in Morrissey's favour in terms of amount contributed to the song.