Learn Dutch: De kinderen glijden door de modder heen. - The children slide through the mud.

While door de modder simply means "through the mud", adding heen creates the circumposition (a two-part preposition) door ... heen. This strongly emphasizes the path or passage from one side to the other. It paints a more vivid picture of the children sliding all the way through the mud patch, rather than just sliding around inside it.

Where exactly does heen go if I add more words to the sentence?

heen normally sticks to the end of the prepositional phrase it completes, right after the noun. For example, if you add a time word like vandaag (today), the sentence becomes: De kinderen glijden vandaag door de modder heen. It acts almost like a bookend for the mud.

If I change this to the present perfect tense ("have slid"), where does heen go?

In the perfect tense, the past participle goes to the very end of the sentence, and heen will sit right before it. The sentence would become: De kinderen zijn door de modder heen gegleden. (Note that verbs of motion from point A to B use zijn rather than hebben!).

Is the verb glijden regular or irregular?