Adverb clause: his delight evident

It is an example of a [reduced] absolute phrase.

The following from grammar.ccc.com:

ABSOLUTE PHRASE

Usually (but not always, as we shall see), an absolute phrase (also called a nominative absolute) is a group of words consisting of a noun or pronoun and a participle as well as any related modifiers. Absolute phrases do not directly connect to or modify any specific word in the rest of the sentence; instead, they modify the entire sentence, adding information. They are always treated as parenthetical elements and are set off from the rest of the sentence with a comma or a pair of commas (sometimes by a dash or pair of dashes). Notice that absolute phrases contain a subject (which is often modified by a participle), but not a true finite verb.

Their reputation as winners secured by victory, the New York Liberty charged into the semifinals.

The season nearly finished, Rebecca Lobo and Sophie Witherspoon emerged as true leaders.

The two superstars signed autographs into the night, their faces beaming happily.

When the participle of an absolute phrase is a form of to be, such as being or having been, the participle is often left out but understood.

The season [being] over, they were mobbed by fans in Times Square.

[Having been] Stars all their adult lives, they seemed used to the attention.

It is a participle phrase, with the verb implied. The sentence could be written

He ran over to me, his delight [being] evident, and hugged me already.

The phrase is used adjectivally to modify the pronoun He.

It is very common to omit the participle being when the phrase includes both a noun (such as delight) and predicate adjective (such as evident).