When I click on Acropolis, the top bar says "Send" to Acropolis, not to City . But when I send defensive pieces  to provide defense  by sending pieces from my acropolis they automatically return to my City.  How can I directly send pieces to city without raiding or providing defense for  someone?

To recall from Acropols(to City) - Press Recall in Acropolis :)

To send (from City) to Acropolis - Press Send in Acropolis

if you want to send troops from your city to acropolis then click on the send button in acropolis or open war council and send from there and if you want to send troops from acropolis to city click on the recall button