In my situation, I would try to start my 03 Golf TDI in the morning; and it would just crank and crank and crank. Then after about 30 seconds or so, it would finally, but reluctantly start, hardly run; but then run fine after a while.
So after that, it would run fine; and it would also start again easily after it warmed up.
At first, I though it was the glow plugs; so I checked them with the ohm meter. GPs were okay.
Then I checked the glow plug harness, the GP fuse, tested the relay, etc. etc.
I almost gave up, but then after finally getting it started, I was staring under the hood and thinking what could be wrong - all this while the car was running.
Then I saw a tiny stream of bubbles in the incoming fuel line.
After I tightened all the fuel connections, it started right up the next cold morning. My problem got solved.
But to answer your question: Yes, the air leak was happening all day. The TDI can take an air leak, but only after it runs a while.
Apparently, the fuel injector pump needs to for full for a fast start.