Problem with my car jumping?

I have a problem with my car

The car is a 6 year old Clio 50k miles 1.2 engine, manual trans. It acts perfectly fine when cold, but after about 5-10 minutes of running it, it sputters and bucks, If you accelerate VERY slowly, it is not so bad, it feels as if when jumping it isnt accelerating as very hard to gain speed. Another unrelated problem is that it grinds when shifting into 1st gear. My primary concern is the bucking/jumping. Any ideas would be greatly appreciated.

will take it to the garage this weekend but if any of you know what it might be would be really grateful. Some people have said it might be either, Air flow pipe, Fuel ignition/pump or spark plugs but no one is certain.
Just a note my car temperature reader is out by like 30 degrees always makes it around -30 so annoying been like this for around 4-5 months not sure if related?