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Michael pointed out few things regarding my site - in the end, problem started with Joomla 3.8.12 (but I obviously didn't noticed it), and updating it to 3.8.13 and 3.9.1. didn't help.
Now, my site uses Joomla 3.8.10, it is working fine, have one small issue (not affecting functionality), just have to find the way how to upgrade it to the latest joomla version, without causing issues ajax component.