Q:

what is the effect of a current-carrying wire through a solenoid?

The solenoid coil concentrates the magnetic field lines through the interior when the solenoid wire carries a current.
I am curious about a different case, however. If a (separate) straight wire went through the center of an unpowered solenoid coil, and this wire carried current, what would the induced effect be on the solenoid coil (say the solenoid is either grounded or left open)?  Would any induced effect fight the current on the wire through the center?

A:

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

