He has come a minute ago.
He came a minute ago.
I would usually want more information (this above sentence is grammatical but sounds incomplete) - He came here a minute ago.
I have had my breakfast a few moments ago.
I had my breakfast a few moments ago.
The train has left the station not long ago.
The train left the station not long ago.
Do we use time markers like "a minute ago", "a few minutes ago", "not long ago" with present perfect?
I'm not too good with tenses, sorry, so I have to intuit this, but from the above corrections, it appears not. I believe you would usually just use the past simple.
And what is the difference between "a few MOMENTS ago" and "a few MINUTES ago" and are these definite time markers?
As far as I know there is no difference from a grammatical point of view.
If so why can we use these with present perfect tense?
As far as I know you can't.
And could we use "He has left just a few moments ago or a few minutes ago."?
You would use "He left just a few moments/few minutes ago."