Thank you for your help. I purchased the SafetyGate™ Professional Retrofit Restart Protection Electrical Plug. It was just what I was looking for. It works great. Thanks again, I appreciate your help.I have to concur with the advise to contact a different service company. Foot valves can be replaced, though it's not recommended for a do-it-yourselfer because of the risk of dropping the pipe into the well -- and if that happens, you might be forced to drill a new well. Keep in mind that well pipe full of water can be very heavy*. On the other hand, if you have a driven well (or a small-bore drilled well) where the suction pipe also serves as the casing) then a new well may very well be necessary.
You don't need a relay system. All you need is to replace the existing pressure switch with one that has a low-pressure cutoff: if the water pressure drops below a certain point, the pump will shut off (or not turn on). These switches have a control lever on the side; to restart the pump after it loses pressure, hold the switch in the "start" position until enough pressure is built for it to maintain "on."
On second thought, a relay might be easier to install if your pump is connected to a 110-120 volt outlet. This device is marketed to prevent power tools from restarting, but should work for your application:
Neither of those solutions solves the problem of a leaky foot valve, which wastes electricity and puts unnecessary wear on your pump. As for the foot valve: you want to make sure that's the problem and that there's not a leak somewhere else. If you can shut off the water to the house & yard, monitor the pressure gauge. If it goes down, the leak is upstream of the main valve (between the foot valve and pump, or between the pump and the main valve). If it does not go down, then you likely have a leak elsewhere.
*In a 1-1/4" nominal diameter pipe, the most common size for single-pipe shallow-well above-ground pumps, there are 18 cubic inches of water per linear foot. That's about 0.65 pounds. Steel pipe weighs about 2.23 pounds/foot. Add those together, you get about 2.78 pounds/foot. Multiply that by the depth of your well (to the foot valve) and you'll get an approximation of the weight. So for a 20' depth, you're looking at around 55-60 pounds. On the other hand, if you have a submersible pump that's 300' down, you're looking at around 830 pounds, plus the weight of the pump and wire!