I used a product named: Snow Seal. Your pictures look like a Henry's product. Wet patch is black.

Tar would be difficult for a DIY

How old is the roof and why is it failing? Is the membrane cracking or is it just opening at the seams? I'd be calling for estimates on a new roof, even if I did a DIY patch in the mean time.

JM$.02W

Much appreciated for both comments. To be honest, I'm not sure how old it is, we recently purchased the house. It doesn't look like it's at a seam, It looks like the membrane is cracking

" Is there any reason I should be using an experienced roofer, rather than doing this myself?"

The fact you don't know what's going on is reason enough to call a pro. One can do more damage than good if they don't know what they are doing. Sealing epdm with asphalt base sealant is one common way of ruining an expensive roof. The only thing flat roofs have in common with shingle roofs is both cover a building. For that reason,use a company that does only flat roofs. You will find them under commercial roofing. When I first started overseeing commercial buildings I thought all my employer's buildings looked to need new roofs judging from appearance. Turned out most were in great condition,others only needed maintenance and all had several years use remaining. You are better off giving yourself a hair cut than experimenting with roofing.