The problem in this particular case was that the troubled areas were not always seen easily and there might be too many of those to delete them manually. Below is the process I've used, may be not too elegant, but worked well:
1) Within a file geodatabase create a feature dataset with the same spatial reference as the polygon feature class and insert the FC into the feature dataset;
2) Create Topology within the same feature dataset, adding the Must Not Have Gaps rule;
3) Validate the topology by right clicking on it within Catalog Window, then add to the ArcMap Table of Contents (TOC);
4) Start Editing session and turn on the Topology toolbar;
5) From the Topology toolbar, open the Error Inspector and search for errors in violation of the aforementioned rule;
6) The portions of the polygon boundaries forming the loops must be identified;
7) Select all those features in the Error Inspector holding the CTRL key, then right-click on one of them and select 'Created Features' - the new features will be added to the polygon feature class, each corresponding to a particular 'loop';
😎 Select all the 'loop' features that you want to get rid of along with the polygon they belong to. From the Editing drop-down menu select 'Merge', and identify the polygon which you want the loops to be merged to - the 'loop' features become part of the polygon, therefore eliminating the gaps formed by extra vertices.