Why is my mosfet driver circuit clipping lower voltages?
The Mosfet needs at least a couple of volts on the gate before it begins to conduct anything meaningful and the signal from the op-amp will drop below this figure and turn the FET off.

One thing that you can try is connecting the feedback resistor from the source instead of the op-amp output. Unlike your previous question, in this configuration, the FET has no gain - it's just a voltage follower so it won't oscillate.

By reconnecting the feedback resistor you are ensuring "supply meets demand" where you need it - on the source.

If you have any troubles it will be because the op-amp is not a "rail-to-rail" output. For instance with your current circuit, if you measured the output from the OP-amp directly you'd see that it clips because it can't drive it's output transistors all the way to ground or the positive rail. You might get away with it though in my proposed circuit.