If you are seeing 10V at the battery with it running then you have other issues.
I would add some other tests to the list.
Hook the positive meter lead to the alternator and check the voltage at various grounds.
- I would want to see what voltage it was showing at the body side of the transmission grounding strap.
- I would want to see the voltage at the ground stud where the negative battery cable mounts plus at the actual negative post on the battery.
If you have a huge voltage loss somewhere down that chain you have an issue to work on.
Same basic test on the positive side. Hook the meter to the ground right at the alternator.
- Check the positive voltage at the starter.
- Then check the positive voltage at the positive cable at the battery and then at the actual post.
If you see a huge drop along a cable the size of a starter cable then the cable path has issues.
In all my years working on cars I have seen 10 volts on a running car at the battery but never with it showing 13.8 at the same time at the alternator. There is WAY too much conductor between the two to have that much drop without a lot of resistance in the path.
Someone with an electrical engineering degree could probably calculate how much load would be necessary to get a 4V drop at the end of a 10' battery cable but it would be more like an arc welder then a battery.
I found a voltage drop calculator and put a 10 amp load through an 8 gauge wire 10 feet long and the drop was 0.13V. Now I'm assuming the positive lead is about 8 gauge but it is nowhere near 10' long.
Something else is wrong...
Or the 10V is when the car isn't running.