Can we make a PDA with only 1 state which accepts a^n and n is odd?
But can we have a non empty stack at the start?


By the formal definition PDA has the initial stack symbol $Z$ with which a computation starts.

The idea is to use three stack symbols $Z, Z_{even}, Z_{odd}$ to keep track of whether the number of $a$s read so far is even or odd. If the PDA reaches the end of string and topmost symbol is $Z_{odd}$ then empty the stack.