1991 Mazda 626 Handbrake light

The hand brake light should not come on at any time during braking, hard or soft. Does the handbrake light go on and off with normal operation of the hand brake? Although unrelated to the hand brake light, but perhaps relevant to your problem, have you checked the brake fluid level in your master cylinder? Have you had your rear brakes inspected recently? Has the peddle pressure or braking performance degraded in any way?

The handbrake works fine, and the light on the dash will come on when the handbrake is applied. But when I brake firmly the light also comes on, then will go off when I release the brake. I have not noticed anything different with the brakes, however I have only been driving the car for a few months, so I am not very familiar with it. I have checked the brake fluid level and it is fine. Before it took its MOT in January I changed the front left CV joint, wondered if the ABS might be causing it. If I was to check the back brakes, what would I be looking out for?

Let me verify that when you pull up on the parking brake, the parking brake icon on the dash lights up. And when you step on the brake pedal, the same parking icon lights up, and NOT a separate brake light. I would inspect the rear brakes to verify that the parking brake is operating properly, is not activated when you step on the brake pedal, and to diagnose whether your problem is mechanical or electrical (light-related issue only). I can't remember if your car has disc or drum; if drum, you'll want to remove the drum so you can inspect hardware and ebrake connections. You'll have to be very, very careful when pressing the pedal down on a drum system as you can pop the wheel cylinder pistons out. Even if you don't have experience with this test and don't want to try it, look for shiny wear areas on the ebrake cable, housing etc. both inside and outside the braking area. A mechanic's mirror is handy so you can look on the back side of springs etc. If all is in order, you likely have an electrical issue. If you want to track this down--you'll need a factory-quality wiring schematic to trace common power and ground points between the brake pedal circuit and the parking brake. Problem could be in the instrument cluster itself.