Hmm, I think that to compare that voltage I would have to supply the comparator with higher voltage than I actually want to measure. That would force me to use boost step-up converter. That leads to reduction in simplicity, but I would do that if there is no simpler solution.

Doesn't need to be higher, comparator can work with both higher and lower voltage levels than reference voltage or the other way. Efficiency will go down if use the same supply or a battery powered device and use inverter if output needed is of opposite polarity.

 

Houston - we have a problem!
The LM393 is a comparator with an open collector output. Normally it is an open loop digital device, but the feedback resistor provides hysteresis.
The LM358 is an operational amplifier, and a pullup on the output is....ah unusual to say the least.
I would not expect this to either work or provide useful information.
Something is not right with your connections if you can move a node in a circuit around my measuring or touching it, then there is no driving source and you have nothing but high impedance paths. If you have a low impedance driving source you cannot do this.

Hmmm, my hystheresis resistor R8 has high resistance (680k), maybe too high so that it interferes with multimeter internal impedance?

Anyway, I tried removing TC4427 from the board and bridging its input and output. After doing so, the LM393 is responsible for turning on/off the transistor. To my surprise, device is working good in this configuration!

 

Hmmm, my hystheresis resistor R8 has high resistance (680k), maybe too high so that it interferes with multimeter internal impedance?

Anyway, I tried removing TC4427 from the board and bridging its input and output. After doing so, the LM393 is responsible for turning on/off the transistor. To my surprise, device is working good in this configuration!

The most likely explanation is that you managed to resolder the faulty joint when you removed the TC4427

 

The most likely explanation is that you managed to resolder the faulty joint when you removed the TC4427

Hmm, I don't think so since I'm using this DIP8 socket.

Whenever I plug the TC4427, the problem appears :/

 

Ok. I decided that I want to change TC4427 to some other ic, or just transistor circuit. Will push-pull circuit be suitable? What do you suggest?

 

It's not possible to recommend anything, as the TC4427 clearly isn't the problem,.

 

I checked my connections on the board twice, and I everything is fine. No loose conections. What else can cause this problem?