I found answer for my question here.

I was not sure how to make precision link to particular posting, so I pasted it below as is:

"Quote Originally Posted by DrByte View Post

I realize you asked specifically about the Mobile Menu, so I will assume that you've already turned off Category Counts here: https://www.zen-cart.com/content.php?198

And, doing that DOES turn off the Category Counts, even in the mobile menu.
But you're probably referring to the "sub-menu counter" that gives an indicator about how many sub-links are available in each sub-level in the mobile-specific menu.

So, to turn off the counts of how many sub-levels are available in the whole mobile-menu, edit /includes/templates/YOUR_RESPONSIVE_TEMPLATE/templates/tpl_modules_mobile_menu.php:
Line 159:

Code:
counters: true

... change that to false, and ALL sub-level indicators will no longer have ANY counters. Be sure you assess whether you really want that turned off"

I tested it, it works as described. It fixed the same issue in Tablet mode as well.

Thank you everybody for the effort to help.