Is the question about the difference between 'Can I ...' & 'Could I ...' and 'May I ...' & *Might I ...'?
I will not address the difference between can and may as this has been discussed in the comments on the question (they are now part of idiomatic English for better or worse).
May and might are known as modal auxiliary verbs. According to the Oxford Guide to English Grammar, 'with modal verbs, we express ideas such as actions being possible or necessary' (p. 78) and 'the modal verbs are will, would, shall, should, can, could, may, might, must, need, ought to, and dare' (p. 113). In most cases, modal auxiliary verbs do not have tenses, and The Cambridge Guide to English Usage asserts that 'May I have a cup of tea?' and 'Might I have a cup of tea?' are grammatically the same, with the second '[seeming] overanxious to let the other party determine your right to the simplest of drinks' (p. 340).
In short: 'Can I ...' & 'Could I ...' are part of idiomatic English, but I wouldn't recommend them. The difference between 'May I ...' & 'Might I ...' is not grammatical but stylistic. All things considered, it is probably better to use 'May I ...' rather than any of the other alternatives as it is definitely correct and does not show overanxiousness.