In a cover letter (for a job application), is it correct to start the letter with "By this letter,…" (and then introduce the purpose of the letter)? this sounds a little strange to me, but there are some example of it on the internet. So, I would love to know whether this is correct, and if not, if there is a similar but more correct way of referring to the letter (as a form of gentle introduction) while introducing its purpose.
The usual way to convey this meaning is to write "I am writing to ... (inform you/tell you) that...". I've never seen the expression you cite, not in English. I know it is accepted in other languages, maybe non-native English speakers translate it from their mother tongue in English.
By this letter or with this letter does not add to the message you are conveying. It can simply be omitted or preserved to emphasise the fact that the letter being received serves the intended purpose.