Oh dear, you don't seem to be having much luck with this car.
If it is the well-known and unfortunately all too common flap actuator fault, then if you want to repair it properly, the parts cost over £1000 (you can only buy the heater box complete from FIAT and sourcing one secondhand is a huge risk, given how fragile they are) and the labour to fit will likely be the same again, since most of the interior forward of the seats has to come out, including the steering column and both doors.
There are workarounds, but if done incorrectly can make things worse, and if the workaround fails, you don't get a second chance.
If the garage you bought it from will cover this under warranty, then personally that's the route I'd be taking.
The flap actuator is a huge weakness in Fiats with climate control and for me, having climate on a Fiat would be a dealbreaker.