What your professor has to recognize is that an HPLC column is a consumable. It is not meant to last forever. The back-pressure shows that it has been mistreated in the past with careless handling. Also, the HPLC pump seal could be going bad and generating polymeric microparticles which are clogging the column. At both ends of a column you have very fine sieve called frits (micron sized, typically 2 micron pores).
If your solvent, sample had small particles they can easily clog the frits. Also, if someone were careless enough, a sample can precipitate at the head of the column due to solubility issues. It is always the inlet frit which causes problems.
Look at the schematic of a column:
Some standard ways to check the system:
Prime the pump, which just means that you ensure pump is filled with solvents and there are no air bubbles.
Remove the column and connect the injector to the detector with a long tubing. And start the flow, always start at low flow rates like 0.2 mL/min and then slowly change it to 1 mL/min. How much pressure is observed? There is negligible pressure like 5-6 bars, then the problem is in the column.
HPLC columns have an arrow to show the direction of flow. Intentionally use the opposite direction and start the flow, again slowly. Do not connect the column to the detector. Starting at 0.1 mL/min and then go up to 0.5 mL/min. If it still stays at the same high pressure even after an hour, it is time to either toss it in the garbage.
If your professor is more mechanically oriented, open the column (last resort). Sonicate the inlet frit in various solvents and fill in just bare silica if you accidently scraped a layer of stationary phase and put the cleaned frit back.