Which is correct?
(a) I just spoke to Mr. Smith today and I asked him to help us.
(b) I have just spoken to Mr. Smith today and I have asked him to help us.
As far as the tense usage goes, you can use both but with a little modification:
a) I just spoke to Mr Smith and .....
a) I spoke to Mr Smith today and....
b) I have spoken to Mr Smith and.....
If you use the present perfect, then you don't specify when (although you can say "I have spoken to Mr Smith in the last few days....")