My old bike pump psi gage shows a different pressure than my newer hand held tire gauge. Is there a way to fix the pump or do I need a new one?
You're not going to be able to fix something like a pump: you won't be able to buy spare parts for it and, even if you could, you could probably only install them using the manufacturer's tooling.
However, the fact that your pump and pressure gauge disagree doesn't tell you which one is wrong. Maybe your trusty old pump is actually right and your new gauge is inaccurate. Maybe your pump really is old and worn out.
If your two devices each measure in a consistent way (e.g., one always reads 10psi lower than the other), then just learn the translation between them. Assuming you're not running at close to the maximum pressure of your tyres, it doesn't really matter if you inflate to 80psi or 90 psi, as long as the results are comfortable to you. If you're happy with the state of your tyres when you pump them up to a certain pressure as measured by your pump, then keep doing that and just be aware of the different number your pressure gauge will tell you. It doesn't matter if they're not real pounds per square inch (or bar or whatever scale you prefer).
If you want to decide which one is wrong (and it might even be both!), you'll need to use multiple devices. Can you borrow a pump or gauge from somebody else? Your local bike shop, for example, probably has a pump they'll let you use.
If the older pump reading incorrectly compared to the pressure gauge by a small, repeatable percentage or fixed error, you can just translate the incorrect reading.
For instance, of you want to run 70 psi, but the pump reads 65 at the pressure, just pump to 65psi as read on the pump gauge.
As already mentioned the pressure gauges of these pumps show often wrong values. Friends and I have pumps for tubeless tires which are not cheap - and even some of those are quite off.
So we often use these special electronic bicycle tire pressure gauges. E.g. from Topeak or Schwalbe. But I'm not a fan of those. I bent the thin threaded rod of some presta valves with the Topeak and my Schwalbe soon had an electronic issue.
So more important as the absolute value is the pressure you want to have. As long as the value is consistent it should be ok. If you want to know which pump is more accurate you can compare it with other pumps - maybe from friends.