Will a resistor fixed just in front of an op amp change its gain?
Will a resistor fixed just in front of an Op Amp change its gain?

RP is after the op-amp. (Note that op-amp is an abbreviation of operational-amplifier. It's not a proper noun so it doesn't get capital letters.)

However what I think is that since no current flows via an ideal Op Amp hence no current flows via Rp, therefore it's presence changes nothing.

No current flows into the input in an ideal op-amp. An op-amp with no output current would be useless.
RP does change the gain of the op-amp circuit (if you consider the left side of R1 as the input and the output of the op-amp itself as the output). To get the required VO the op-amp now has to &quot;stretch&quot; further because of the potential divider effect of RP and R2. The gain of the circuit is \$ - \frac {R_2}{R_1} \$ at VO but it's \$ - \frac {R_2 + R_P}{R_1} \$ at the output of the op-amp itself.