HP's in general are not known for having even halfass-decent cooling systems most particularly due to their poorly engineered airflow designs, so running it without your fan may have damaged internal components. That being said, the laptop is still running so perhaps you're mostly fine, you'll just be running on borrowed time.
Your processor is throttling due to the heat, which is forcing itr to slow down to protect itself from thermal meltdown. While laptop processors can handle a tremendous amount of heat, heat has a negative impact on electronics, with the thermal expansion from heat and vice versa in varying extremes, your processor may need replacement before long. Luckily for you, on most laptops nowadays you can replace the CPU. So you'll know just how to get to it once you've replaced the fan.
When you replace the fan, I would HIGHLY recommend that you remove the copper heatpipes that come off the CPU(and dedicated GPU if yours has one) and leads to the radiator alongside the fan exhaust. Clean out the radiator thoroughly with compressed air in both directions. Then, clean up the old thermal compound which will likely be dried up and burnt onto the CPU due to the constant heat, and then apply a good aftermarket thermal compound like ICdiamond 7. Put it all back together and your temperatures should be just like they were when new, and your CPU should not be throttling anymore.
You should monitor your temperatures with hwinfo to make sure your temps are good. If you still have high temps, chances are your thermal paste was improperly reapplied.
