When I start the car blue smoke comes out of the exhaust and then the car cuts out after it struggles to idel
Quite correct, unfortunately blue smoke for burning oil, black smoke for rich mixture.Blue smoke on statrup or overrun is usually a sign of oil being burnt.
It could be valve stem oil seals, valve guides or piston rings.
Any of these jobs are fairly big to do and could be costly, especially the piston rings.
I hope i'm wrong, but, as i have found, in the past blue smoke has always ended up being one of the above causes.