Can you send me a live link so I can troubleshoot the issue?

Sure.

Link

After your suggestion I have asked FavThemes support about this issue and here's their answer:

"The template is not designed to limit in any way the number of images and/or taget the first-child ( this means the first item from a list ) and/or last-child ( the last item from a list ) to show or not to show the images or the articles.

Now, K2 also has a very specific way of inserting images inside articles (K2 Items). The code for the article where you have the image has an image block with specific K2 CSS classes that are the default ones from k2.css and are not overriden by k2style.css of the Favourite template: "tagItemImageBlock" and "tagItemImage".

In the code for the article where the image is missing, these classes are absent, like K2 does not even have the image inside that article.

Can you please try to test if you enable one of the default templates like Protostar and Beez if the images will be displayed? If the images do not appear at all, even when using a default Joomla! template, it means that this has to do with the K2 settings and it's not something controlled by the template."

I have tested with protostar template and problem is still existing.

I can see here: mastertec.webd.pl/czytamy/index.php/edukacyjne/itemlist/tag/Ma%C5%82gorzata%20Strza%C5%82kowska
that several items are displayed with images, not just the first and last.

Are you sure these items (with no images) do not have any specific settings ?
Also which other extensions are you using?

Yes, you're right. It was first and last article, but in meantime other users added more articles. It's curious that all images for those new articles are displayed properly.

All settings - for subcategories and articles should be inherited from main category without any overrides on subcategory or aricle level (I have checked to be 100% sure). All of those articles are in one subcategory "edukacyjne".

Other extensions should be safe - I have used them before with K2 and I never had any problems. I have acy mailing, akeeba backup, breezing forms, dm pinboard and jce. There are also some modules: cookies accept, ga k2 slider and news show sp2.

I have used most of these extensions with no issue, not the modules though.

Can you delete the image of an article which is not displaying and try uploading it again?