Which one is better? Are they acceptable or do they sound weird in English?
No. It doesn't matter whether you say help or use;
both sentences are ungrammatical, because of any.
Any in the phrase of any use to you is a negative polarity item (NPI), and therefore it must occur within the scope of a negative trigger. There is no negation in the sentence, so it's ungrammatical.
Both predicate nouns would be grammatical, however, in a sentence with a negative trigger, like
I didn't think I could be (of) any use to you.
I didn't think I could be (of) any help to you.
and in both cases the (of) could be dropped, as well.
Your sentence is fine, although it telegraphs too much begging. Employers do NOT like beggars, nor desperation.
Take charge, and make a confident statement, stating that "I am confident that I am the best fit for the job" or "Please let me know if this opening is available because it's the best fit for my career"
First, it's "cover letter," not "covering letter."
Second, in American English (can't comment on British English in this case), we would never say "I hope I can be of any use to you" or "I hope I can be of any help to you."
Third, Anderson's advice regarding what you might want to write in your cover letter is good advice. You'll need to come up with a lot more than just the couple sentences he recommended, though.
Fourth, there are professionals who help people create cover letters and résumés. You might want to invest in that. It could make the difference between getting that better job or keeping the lame one (or staying unemployed).