It looks as though you have a double lining there; a black, possibly non-woven, layer and a white knit layer. To get to the inside of the finger you will need to unpick the linings - at a side seam? Then if you have some thin enough leather apply a patch on the inside using glue and stab stitch as
@HenriettaMaria described.
This query has prompted me to look at my own leather gloves because I thought the linings were shredding. Once I got them turned inside out I found that the fingers are not fully lined although the thumb is. Basically the lining, a non-fraying interlock knit, is cut to the same shape as the back and front but there are no between-finger gussets, just two strips of fabric for each finger sewn together at the tips catching in the seam allowance of the leather. Two fingers in one glove and one finger in the other had become unstitched. What I thought might be a very tricky repair turned out to be very easy. The fiddliest bit was getting the fingers turned inside out and then back again!
Whilst I've never come across this before I have to say that I wasn't aware of anything unusual in the wearing of those gloves which are three or four years old. I've just had a look at my other leather pair and they have full linings.