The -va/ve ending has two roles. It forms the gerund or present participle and the adverbial participle.

The latter, the adverbial participle, is covered here.

When we need to modify the verb with another verb, the adverbial participle is used. The second, modifying verb's stem has -va/ve added to it, and now becomes something like an adverb.

For example, in this sentence: he came home crying, the underlined participle behaves a bit like a verb and a bit like an adverb. It describes how he came home, so it's like an adverb; but it clearly ends in -ing and so must have a verbal basis.

We can understand how these participles behave a little more like adverbs by forcing the -ly on the end: he came home cryingly.

This is totally articifical, but it may help, particularly in such sentences as he came home hurridly, which is really a Hunglish was of saying "he hurried home."

Here are some more examples. I am putting the Hunglish "-ly" at the end of the adverbial participle that I am demonstrating.

• sietve jött he came hurriedly
• mosolyogva mentem el I went smiling(ly)
• sírva jött she came home crying(ly)
• könyörögve nézett rám he looked at me pleadingly
• nevetve mentem be az iskolába I went to school laughing(ly)
• elkeseredve ültem le tanulni bitterly I sat down to study
• énekelve lépett be a szobába she stepped into the room singing(ly)

When we use the adverbial participle as a predicative adjective, we must not omit van.

• az ajtó nyitva van the door is open
• az ablak zárva van the window is closed
• a szobám fűtve van my room is heated

Temporarily closed

The root verbs in the above are, of course:

• siet hurry
• mosolyog smile
• sír cry
• könyörög plead
• nevet laugh
• elkesered|ik grow bitter, dispair
• énekel sing
• nyit open
• zár close
• fűt boil, heat

There exists an alternative form of the suffix: -ván -vén. These participles cannot used predicatively, they may form implicit sentences, though their form is not modern. For example:

• bemenvén a terembe, leültem having entered the room, I sat down

My thanks to Ádám Czinege pointing out this usage