1) The power amp is prone to high frequency oscillations which can be improved with the 4.7 ohm 0.1 uF snubber load as they show on most audio power amps including this. (which you neglected)
2) The MOSFET is a miss-fit for this application. It is rated for low current and 3 Ohms RdsOn at Vgs=10 but if you only supply a couple volts less than Vcc=5 V voltage so it could be >> 10 ohms and not have a low Vds when enabled. Remember this. If the MOSFET Vgs "threshold" is given use at least 3x the max Vgs(th) threshold. not 3~4V drive for Vgs(th)=2 to 4V range.
3) TIP122 Darlington has pretty high gain which drops when saturated but Vce(sat) starts at 1V and rises to 1.2 at a few 00' mA. and then only needs <1mA base current so you can drive this without a Power Amp (U1) just using 12V on an Op Amp and 12V on the MOSFET switch.
That being said , the FET must be much lower than 10 ohms for R sense, if you want precision current control. Such as 100 mohms. Then all of the lost power from voltage drop in the FET will be add to voltage drop in the TIP122 which naturally increases its temperature unless you designed a heatsink . Basic Ohm's Law Pd=VI from the drop voltage and current selected.
You can easily eliminate the FET with inserting a large series R>
1K from pot wiper then switch Gnd to Vin(-) to disable current.
I forgot to mention in a standard VC-CC sink or active load, you can use a current sense R of only 1/2W (derated by 50%) by choosing a Vdrop at max current of I^2*R = 250mW. Thus if max current is say 0.5A then R=0.25W/0.5^2= 1 ohm .
Thus the max Vref on your pot is now only 0.5V to match 0.5 A * 1 ohm. So you add a series fixed R to the pot to V+ to reduce the control range. Now you eliminate the FET and disable as above using a switch to ground on Vin(+) to ground.
IC's use this approach but use a standard Vref of 75 mV to 100mV for even less power dissipation instead of a 500mV drop. Then all the voltage drop is in your low RdsOn FET or Darlington.
Then if you are choosing large currents like many Amps, then you use a dummy load like a halogen bulb of much larger power rating to dissipate most of the heat, as these tend to be constant current sinks in the 25'C to 500'C range with a large PTC coefficient but still use the above to fine tune and regulate CC with an emitter or drain sense resistor in milliohms.