Very slight leaking roof

We have a very small damp patch in our main bathroom ceiling just started. So had to investigate what the problem is. The area on the roof is in blue in the photo below, (Scaffolding no longer there)

Actually I didn’t realise her was even a gully up there,( can 't check the other side but no problems so far).
I needed to check it properly, to see and get to it with the skyvac hose and tubes I had to get out a very tall pair of steps

This was the only way I could get close enough for the right angle to clear it. The amount of “muck” taken out, well far more than I expected. So now I can see the problem and no doubt you can as well.

A piece missing to start with let alone the lower bit of lead seems to be over the bottom of the gully not underneath and a right bodge up on the corner edge, no doubt done by the builders when the house was built and very little inspection before signing the building off
Already had one quote to put it right, just waiting for another now

Oh if wondering about the photo, I took that way back nearly where the house photo was taken and zoomed in using a compact Panasonic Lumix DC-TZ95 camera

We had a similar problem several years ago with leaks in our conservatory. We got a chap in who specialises in conservatories and he discovered that when the conservatory was added to the house (well before we bought it) they had simply left in place the original plastic guttering of the house roof and which was completely inadequate.

He replaced that with lead but, because the replacement guttering would have to be L-shaped and quite long, he had to use five separate lengths, overlapping the ends where they join. That has been much better although I must go up there once a year, usually in the Autumn, to paint some Flexacryl over the joints because due to expansion in the Summer and contraction when the weather cools, the joints can begin to leak.

As long as I do that we’re pretty much OK, except that the lead guttering is (necessarily) not quite as deep as I would have wished - he couldn’t make it any deeper because of the space between the slates and the wooden structure of the conservatory. Consequently, if there is an unusually heavy downpour, we occasionally find a few drips but nothing too serious.

Looking at the lead in your picture (apart from the gap which you mentioned) that also looks not to be very deep and, depending on the fall, I wonder if that might also contribute to the problem, again, in a very heavy downpour.