Why not I saw him lately?
IMHO "lately" rather means "останнім часом".
"I recently have seen him/
I recently saw him"
seem to be acceptable to me
"I recently have seen him" -> sounds awkward to me. Sounds like it's an unfinished sentence: "I recently have seen him going to..." (still awkward but less)
I recently saw him -> is accepted
Anyone else can comment on this?
You could also say "I HAVE recently seen him" ( this sounds almost too formal or old fashioned?) instead of "I recently have seen him doing that..." but I would agree that "I saw him recently" seems more common and natural.
"Бачила" is past tense feminine. So should it be "бачив його."
The speaker is a girl, so she haven't seen him for a long time