What can I use to fill a 5mm gap between a PVC window frame & a double-glazed window pane that it too thin?
There may be a possibility of finding thicker glazing strips to hold the glass in tighter. Window manufacturers sometimes use thinner glazed units in other windows they make, so if you can find the producer of the unit, you may find a glazing strip to cut to length that will hold it in. That is the better of the options I will suggest, although it may be really difficult to get it.

The other is to, I will write this with my head down and my eyes not meeting yours, metaphorically speaking, cause I think it is really a second rate way to fix it.
That is, use silicone caulk to back the glazing to hold it to the seal on the outside. The biggest problem is the silicone will not press the glass against the seal to do the job it needs to keep the rain out. It will be messy if you are not familiar with caulking, especially with silicone, and the window may still leak because the pressure is not being applied to seal the window at the outside face. Laying the window down flat so the weight of the window is pressing the seal, then apply the caulk, let it cure, then set the window sash back in place in its jamb.