Is it possible for this sentence to mean 'I prefer to feed the dog'? If not, how would you say that?
See the notes for the section 'Prefer':
I recommend that you re-read the tips and notes for the "Prefer" skill ( https://www.duolingo.com/skill/cy/Prefer ) -- it shows the difference between the "I had better" and "I prefer" constructions.
Curiously, this seems to be the one sentence practising this construction that I got in the whole skill; the sole exception being for an incorrect answer in a multiple-choice version of this sentence where it said Mae'n well i fi fwydo'r gath.