air leak will cause all sorts of idle issues,

increasing fuel pressure will mask the fault as it will run rich, just like it does on cold start when you say it is ok.

pressure test the inlet from the turbo up.

set the base idle when hot with the iscv unplugged via the base screw under the throttle body.

check fuel presure is 3.5 bar with the reference pipe off and blocked,

is it on p8?