here �s a very good one with examples of usage
as to your question:
by correspondence - that is to say "using letters"
E.g. We kept in touch by correspondence.
never heard "by one �s watch"
I did hear "by the clock" which means at the exact periods of time
and "on one �s watch" means when someone is watching, E.g. That won �t happen on my watch!(i.e. when I am keeping a close eye on what you �re doing)
At a guess = aproximately/ roughly speaking/ I assume that
there �s no such idiom as "at the prospect" it feels like it �s out of context...
It could be part of the sentence
See the meaning and usage of prospect
here
Sophia