I wish he would write more often...

[‘wish’ and ‘would’ cannot have the same subject. We cannot therefore have ‘I wish + I would’:

I wish I would write more often. (Incorrect)

I wish he would write more often. (Correct)]

Please explain, it is correct?

I agree.
“We use ‘would’ with ‘wish’ in a little bit of a special way. It’s generally used about other people who are doing (or not doing) something that we don’t like and we want that person to change.
 I wish he would stop smoking.
 I wish you would go away.
 It’s strange to use ‘wish’ + ‘would’ about yourself, as you can change your behaviour if you don’t like it!”

When writing about yourself you would use ‘I wish’ + ‘I could’.

Rita,

You can of course say - I wish I would - in exceptional circumstances when you are criticising yourself about doing something you can’t help doing or that you do automatically - I wish I would stop using that expression but I say it automically without thinking.