Increase number of people on time after a month (7)
I believe the answer is:
augment
'increase number of' is the definition.
(I've seen this before)
'people on time after a month' is the wordplay.
'people' becomes 'men' (men are people).
'on' says to put letters next to each other (in a down clue, letters appear on others).
'time' becomes 't' (abbreviation).
'after' says to put letters next to each other.
'a month' becomes 'Aug' (abbreviation for August).
'men' next to 't' is 'ment'.
'ment' after 'aug' is 'AUGMENT'.
(Other definitions for augment that I've seen before include "Make larger" , "Make greater in size, number, etc." , "Supplement" , "Enlarge or increase" , "Swell" .)