genitive absolute

A noun and a participle in the genitive case and not modifying anything else in the sentence is a “genitive absolute”. You will translate genitive absolutes with a dependent clause. If a noun is supplied, then that noun will be the subject of the dependent clause (even though it is genitive). The participle will become the main verb.
For the DMW, you need to know the tense of the participle.

If present, use “while” or “because”;
If aorist, use “after”.

present tense And the sea was aroused because a great wind was blowing. ἥ τε θάλασσα ἀνέμου μεγάλου πνέοντος διεγείρετο. (John 6:18)
aorist tense But other boats came from Tiberias near the place where they ate the bread after the Lord had given thanks. ἀλλὰ ἦλθεν πλοῖα ἐκ Τιβεριάδος ἐγγὺς τοῦ τόπου ὅπου ἔφαγον τὸν ἄρτον εὐχαριστήσαντος τοῦ κυρίου. (John 6:23)