So my mother has a leak in her roof (flatish) and doesnt have the money to pay for replacing/repair so thought I'd try roof help out. Here are pictures of the roof and a spot that may be where water is coming in. Am I able to use black roofing cement with a mesh to help seal that or does the roof not seem to be in the best condition overall? Any thoughts or advice would be helpful.

She had the rest of the roof with shingles replaced last year but this part wasn't touched. This section of the house was an extension added on over 30 years ago. No idea when the roof was last touched here.