Brand of kitchen sink with crown logo and CSA stamp

There should be a couple of threaded rods that are screwed into the body of the tap and stick out about an inch, pointing the same direction as the hoses. There should then be a rubber washer and a steel washer in a horseshoe shape with a couple of holes in them that slide onto the threaded rods from underneath the sink. When you tighten a nut on each of the rods, the washer clamps the tap onto the sink.

If yours are missing, you might search for a "monobloc tap fixing kit". Here's an example for illustration but you'll need to make sure you get one that fits your tap.

I'm going to assume you are 100000% correct. But this is an assumption. The only way you are going to know that this is the issue is to hire a plumber, or clear the issue yourself.

This said, your concern of "pushing it further" is real, but realize that millions of people use or eat fatty substances in their home cooking and never have these issues. this leads one to suspect your drain was already nearly clogged for any number of reasons, or someone poured a nice plate of ham (bacon etc) grease in there.

Either way - speculation aside...

Just get a bucket, remove the elbow and look inside and see..... You can just replace the trap at this point. Make sure there are no rough edges for food, hair, etc to catch on. Chances are a toothpick or something is in there and this is building up around it.

Generally, small amounts at any given time will make its way safely to your sewer and you'd never be the wiser. Large amounts, or some other clog may have exacerbated this issue.

Decide if its bad enough to have your main line snaked for extra prevention, but chances are you do not need to do this.