1. I saw my old shirt, hanging on a nail, was crying.

"hanging on a nail" is realtive clause or someting else.

It is poosible to write it as
2.I saw my old shirt that was hanging on a nail was crying.

is (2) is correct?

If we expand your original sentence to 'I saw that my old shirt, hanging on a nail, was crying,' then we can analyze it more clearly:

'Hanging on a nail' is a nonfinite clause, and can be represented by 'I saw that my old shirt, which was hanging on a nail, was crying.' This is a nonrestrictive clause modifying 'shirt'. (Your 'that'-clause is a restrictive one.)

What does it mean to say that your shirt is crying?

Or are you deliberately trying to make a sentence that is correct grammatically but has no sensible meaning?

Best wishes,

Clive (whose sweater is puzzled by this)
Students: We have free audio pronunciation exercises.
what is the wrong construction in the sentence?
 Clive's reply was promoted to an answer.
Teachers: We supply a list of EFL job vacancies