it appears to be wet on the inside, so I expect it's got a leak. I can't see where it's coming from. is there any sign of damage to the grey felt or porous membrane?
Water tends to run downhill, so look above the wet patch.
I can't see moss causing blockages on the tiles.
I think the wet patch is above the level of the gutter, so probably not there. Run some water into it with a hose and see if water tracks inside. If not, try it on the tiles above, and then squirt the verge (the mortar filling showing on the edge of the gable) which looks in poor condition, but is not difficult to renew. I am not a roofer but it probably needs new undercloak and mortar fill. The old must be ripped out, not patched. Not difficult. There are plastic "dry verge" systems which I think are better, but hard to add to an old roof properly unless it is being stripped and renewed.
See which way the prevailing wind blows in case it is blowing rain inside (not likely).
Ask around for personal recommendations of a well-established local roofer. They will be quite pleased to get a job near the ground.
There is a faint chance it could be condensation if the wall by the patch is wet, e.g. from leaking gutter, but I think it unlikely.
Presumably it gets worse after rain?