I am not sure what to do. ie dig deep enough to get below the clay to allow water to absorb back in to the earth?!
Check with a Contractor before implementing this, use the spring to water your lawn, flowers, garden etc. if possible
First determine if the water is coming from a well head line or the city water supply line. You can usually get a good understanding to if this may be the answer by seeing where the water supply comes into the house. This is usually near the house water shut-off valve. Align the house supply location to either the well head or the street supply shut-off valve location. If this new ground water is within a moderately straight line then it would be a good assumption that you have a busted pipe or fitting.
The good news is should be an easy fix. The bad news is someone needs to dig down to find it. The last one we did was the connection brass nipple at the well head and it ended up being 5’-0” underground. Once we finally dug down and exposed the breach we got the plumber out to do his thing, took him 20 minutes to fix it. I will warn you must hand dig, do not use a machine to do this. Also be careful once you start getting close to the pipe to go slow. With a well you will also have a ground line and an electrical line running within the same location. As you dig down, chase the water, it will probably shift around because it is following the path of least resistance.
If it is a spring as you suggest I have not opinion on this situation and hopefully someone else has an answer for you.