A number of faults can be seen:
The painting was not applied in the correct number of coats. Maybe the wood was not prepped either?
The left hand side of the mitre in pic 2. shows what looks like a repair piece that has previously been inserted.
It would pay you to go around all the fascia and barge boards and test them with a thin screwdriver - look for soft rot below the paint job.
The verge tiles (?) do not come far enough out - this is perhaps allowing water to trickle down and return on to the fascia or barge board.
(FWIW: the underfelt (in pic 2.) has been brought too far out - it should be kept back from the pointed edge.)
The make-up timber pieces at the foot of the barge boards have failed (they usually do) and allowed water in and rot to ensue.
Cutting in a repair piece of fascia is simple enough - you come back a couple of rafter tails and splice in the new piece. Roughly speaking, this is what you would do to splice in a piece of barge board. Depending on how much you intend to use then a sheet of ply can be ripped, but for short pieces then a length of timber might do?
However, i think your woodwork is perhaps beyond a little remedial work.
You might have to disturb the tiles and pointed verge, and drop the guttering - and once repair pieces begin to be too many and too long then its time to consider fascia and barge replacements.
As mentioned above, the black grained PVC might be the way to go.
Note: what i'm saying is speculation based on two limited pics.