SUP repair/hole

Recently purchased a S/H SUP looked in good nick. later found some bubbles and a little water coming out of a small hole on the deck (which initially looked like a paint chip) while in the sun.

Q is should I continue to let it out in the sun to ensure it has completely dried out (seems like there is no water coming out anymore), and/or should I seal it up while it is warm or do it in normal indoor temperature?

It depends on the brand, I have had starboard with a hole bigger than that for 8 years, also naish seem to last with holes, not sure about that one though, good luck.
pump.

I had this problem with 2 second hand boards.
Providing it is a polystyrene core and epoxy it is never going to "fill up" but it will get heavier over time. And ,depending on use, shouldn't deteriorate much.
I live in Queensland which helps with the following. It takes some attention.
You can use the sun to get most of the water out.
Put the board in the sun and look for the seep points.
Mark them with a marker with a "circle" with the crack/hole at the centre . If you use a dry erase , be careful not to rub off. I use permanant and clean it off with grease and wax remover.
Duct tape close all but one.
Put it back in the sun with the first fault at the lowest point and let the sun heat it up. Put something undet the spot to clooect water.
Because it has holes in it , it isn' going to bong.
As soon as the water stops coming out , put it in the shade and repeat the process after it cools down.
A laser thermometer is handy for this to check when it is back to ambient temp.
Once no more water comes out of that fault retape it and move to the next one repeating the above until all faults have been dehydrated.
NOW! reseal all the holes and check for any more that escaped the first discovery step. Do the same with them until all faults are found.
You can now do you taper grinds to re"glass" the faults.
While you are doing the repairs put a goretex breather ( or equivalent ) in the board. If you can't find one, or are happy to remember to open and close a manual ( cheaper) breather you ca go that route.
You have a board with more air/water in it than it was designed for and it now must breathe, or it will bong much more easily, than it would have before.
Finish it to a std, you are happy with and you should have reliable board.
Please tell any prespective buyer that it has been repaired if you sell it.

WEIGH the board.
Look up the spec weight.
Seal that area.

My Sunny King 14x27 carbon was supposed to be 10kg and was 12.8kg... without fins.