I have a golf cart with six - 6 volt batteries. It has no lights and I would like to know if I can install a 6 - volt light utilizing this 6 - volt system.

the cart is wired for 36 volts operation so you would need a 36 volt bulb
or
you could put the wiring for a 12 volt bub across the terminals of 2 batteries so the negative wire will be on a negative terminal of the first battery and a fused positive wire will be on the positive of the second battery
the batteries are connected in series so the sum total of the voltage is the sum of all of the batteries
by selecting 2 batteries in series you are using the sum voltage of 2 batteries

You load tested each battery? Can't be bad if cart runs good.

Check all your batteries with a meter you may have one with a bade cell.

sounds like you have either a relay or computer that is bad, needs to be serviced

Lets number the batteries to simplify... B1-B2 andB3. Connect a cable from B1,negative, to B2 positive. B2 negative to B3 positive. Now you will have 36 volts by using B1 Positive and B3 negative. Also you can get the same voltage using 6 six volt batteries, the result will be a longer lasting charge, aka, drive further. Hope this helped.

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all you have to do is run the positive wire from the lights to the positive post of one battery and the negative wire to the negative post of the same battery. You will be using just one battery to power your lights. Hope this helps

The exact answer depends on what battery voltage you have but this hint will get you going.
Assuming you have 3 12v batteries, just wire to the first battery + - and you'll have 12 v. If you have 6 6v batteries, wire to the first + and the second -. If you have a voltage meter, you can verify the voltage of my suggestion.
This will work for lights and other resistance circuits but not for sensitive items like radios.
Golf cart batteries are series wired to get the higher voltage.
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You said you have 6 batteries. Are they 6 volt or 8 volt? Hint-- 6 volt has three holes in the top and an 8 volt has 4 holes on top of the battery. Battery manufacturers recommend the use of a voltage reducer wired to the whole bank. This take the 36 or 48 volts down to 12 volts and is regulated. this is recommended so as not to stress just 2 batteries. If you have 6 volt batteries, you can use any 2 that are wired in series.