Yesterday one of my classes was practicing ‘can’ and ‘can’t’ for ability. One question was something like ‘Can you sing?’. One student wrote ‘Yes I can very well’. Maybe I shouldn’t have worried about so much (or still be worrying about it now), but there was/is something not quite about this sentence.
‘I can sing very well‘ is a perfect sentence. ‘Yes, I /kæn/. I /kən/ sing very well‘, ‘Yes, I /kæn/. Very well’, ‘Yes, I can , very well’, ‘Yes, I /kən/ sing very well’ and ‘Yes, very well’ are all perfect answers. So is ‘Yes[,] I /kæn/ very well’ correct English (in which case, why does it feel so awkward?) or not (in which case, why not?)? (‘Yes[,] I /kən/ very well’ is definitely wrong’.)
Perhaps there is something about ‘short answers with auxiliaries’ (‘Yes, I am’, ‘Yes, I have’, ‘Yes, I do’, ‘Yes, I can’) which precludes there being anything else immediately following, so there must be a longer or shorter pause in speech and a punctuation mark of some kind in writing.
Certainly ‘Yes I can very well’ is not standard English, but equally certainly is perfectly communicative second-language-learner English.
(As an aside, there is a difference between ‘Yes, I can’ and ‘Yes I can’. ‘Yes, I can’ is the answer to ‘Can you sing?’. ‘Yes I can’ is the response to ‘You can’t sing’ (either non-ability or non-permission).)