My Anet A8 reads as 120ish degrees Celsius, but it is set to 200 degrees. I don't know if it is reading the wrong temperature because it still pushes out plastic when I force it. It will climb up and then go back down. I don't know if the hot end it broke or if the thermistor is broke.
You are sitting on a fire risk!
If you are using the stock/original firmware, you should immediately stop printing. The stock firmware of the Anet A8 has no thermal runaway protection (see also this answer), this means it will keep heating until the thermistor senses 200 °C, even if it cannot do that for some reason or another.
When a hotend temperature does not read the correct value, your thermistor in the hotend may not be making correct contact (as you say that it goes up and down). Please ensure that the thermistor is correctly positioned, it makes good contact and the wires correctly fastened.
Similarly applies to the heater cartridge, which can fall out and causing a fire if not properly fastened. Ensure the heater cartridge is properly positioned and held in the heater block.
Fluctuations in temperature sometimes are induced by a wrongly positioned fan duct (but generally not that much). The reason why this is not the case here is that you can still push the filament through while it reads about 120 °C; this temperature is generally too low to push filament through. Apparently the hotend is still hot enough to push filament through while registering a low temperature.
Without any other information, the highest odds are that the fan is blowing on the hot end. If the temp drops when the fan is on, and rises when the fan is off, then simply re-position the fan nozzle. If that's not the problem, check the thermistor. If that's not it, check the termination (tighten the screws) of the hot-end wires at the control board.
Next, check the control board. If there's any indication that it's the board, one potential solution is to add a MOSFET (see the link). Even if the control board isn't the problem, I would add the MOSFET.