Tries to influence a school boss: keeps at it (7,5)

'tries to influence' is the definition.
I know nothing about this answer so I can't tell whether it can be defined by this definition.

'a school boss keeps at it' is the wordplay.
'school' becomes 'S'.
'boss' becomes 'head' (synonyms).
'keeps' means one lot of letters goes inside another (inserted letters are kept inside).
'at' says to put letters next to each other (I've seen this in other clues).
'it' becomes 'presse' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
's'+'head'='shead'
'a' inserted into 'shead' is 'sahead'.
'sahead' after 'presse' is 'PRESSES AHEAD'.

Can you help me to learn more?