Fan cutting in and out at high speed is classic sign of the resistor failing on the slow setting. This causes the A/C system to over heat/pressure and this brings the high speed fan on until it cools down, then it shuts off as it thinks the slow speed has taken over....
Look through the grill at the front of the car, you should see a silver resistor around the 2 0'clock position behind the fan it's about 75mm x15mm x15mm. If this is present it has almost certainly failed. not expensive to fix and should be upgraded to the gold type, available as a kit from JMA cars for under £20. Unfortunately if you have facelift car it is a bumper off job. ( Mk1's can be done through the front grill
You will need to either disconnect the battery or unplug the fan harness behind the bumper as the electrics to the fan are permanent live. ( Don't forget your radio codes and correct sat nav shutdown procedure if fitted)