mega 2560 r3 only works without one of 22p caps of crystal of atmega16u2

That looks like a bad crystal more than anything else. The 'loading' capacitors are required to make the Crystal 'happy'... the technical reasons (several) are a bit more complicated and require some intimate knowledge of how crystals are matched in circuit to the driving electronics... IMO a lot more technical than the scope of this post.

Doc

Crossroads suggestion has great merit however with small value capacitor values accurate measurements are difficult due to the effects o f the meter leads... if you have the correct value caps replace them. If not then certainly measuring the part value is going to point out gross errors as he mentioned... and probably fix your issue.