what is the effect of a current-carrying wire through a solenoid?
No, a wire fed axially through the centreline of a solenoid would not induce anything into the solenoid coil. The magnetic fields (if both are energized) will be at right angles. There would be no transformer action.

At much higher frequencies there will be circulating eddy currents induced in the solenoid winding (from the straight wire) but these won't be significant below tens or hundreds of kHz