Looking again at the pics there are traces of sealant and pitch above the gutter as if there was a larger bodge piece stuck on that has blown away in the wind or dropped off from old age.
If there's plenty of fall on the bay then pinholed felt wouldn't be a short term problem. If water is ponding on the roof then you'll get enough going through to get under the deck planks (you may find the original lead under that felt as well).
Dripping gutter won't help either- it is amazing how much damage a small drip can cause.
Remedies;
Sort the guttering so it drains properly and doesn't drip. If its plastic, take it off,clean the seals out, make sure they're seated properly and replace any broken or missing clips. Make sure there is enough fall to the side with the downpipe (1:500 is usual)
Short-term you could paint the roof with Acrypol or similar and try getting a piece of Flashband behind/under all the layers in the pictured corner.
Medium term felt doesn't last forever, get it replaced. There are various options (felt again, GRP, lead, EPDM)- whichever it'd be wise to budget for new roof deck and while the job is being done, stuff some loft insulation in the void between deck and ceiling.
Lead done properly would look lovely and last 100 years or so but there's the theft risk and finding someone who knows what they're doing.
Felt is probably cheapest but short life (20 years max) and hot felt work is a chunky fire risk
GRP would probably be better than EPDM (because of the fiddly shape), GRP is DIYable as well, plenty of Youtube videos to show you how