Hi @dabreh ,
It sounds as though you may have two or possibly three problems.
1. The headphone socket in the computer may be loose or faulty.
2. The 1 st pair of headphones may have a faulty plug.
3. The 2nd pair of headphones may also be faulty.
What you need to do is to find a different ''known working OK'' set of headphones and try them first in your phone. If they work OK there try them in you computer. (make sure that the volume setting is not too loud).
If they don't work there try them back in the phone again. If they work there OK again, then there is a problem with the computer.
Next try both your other headphones in the phone again and check whether they work OK, work intermittently or don't work.
If you have a DMM you can use its' Ohmmeter function to test the continuity of the wiring for the speakers in the headphone by placing the meter's test leads between the tip and sleeve (or ground)of the plug and the ring and sleeve (or ground) of the plug.
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What is the make and model number of your computer?