I have problems with my MicroUSB connector. There are problems with connection when I simply plug my charger or usb cable. Only when I stronger push cable connector inside I can use peripherials. I have looked at the inner surface of the microUSB and I noticed that one of the contacts is very dark, almost black. I don't know if this is dirt, rust, patina or something else.I have tried to clean it with toothpick, needle etc., but nothing changed yet. Still dark. Can anybody tell me if my problem is caused by dirt in microUSB or maybe there is some other reason for that. And how to successfully clean microUSB.

Sounds like the pin may have become oxidised. You can get contact cleaners like Deoxit but a can of that may be more expensive than taking it to a local electronics repair shop and asking them to clean the contact.

Make sure you've tried different microUSB cables. I once couldn't get a charger to work. The charger was outputting correctly (I had a USB LED multimeter with me), the phone socket was clean but I'd completely missed the small grain of sand stuck in the end of the cable I was using.