ἀρνύμενος ἥν τε ψυχὴν καὶ νόστον ἑταίρων.
struggling to save his own life and the homecoming of his friends
ἀλλ᾽ οὐδ᾽ ὣς ἑτάρους ἐρρύσατο, ἱέμενός περ:
BUT HE COULD NOT SAVE THEM, THOUGH HE TRIED...
["Yes, I spend my men like money..." --"The Sirens" by jh]
For comparison, see Augustus Taber Murray's translation (1924), which also mentions Ares:
 And they that held Argissa, and dwelt in Gyrtone, Orthe, and Elone, and the white city of Oloösson, these again had as leader Polypoetes, staunch in fight, son of Peirithous, whom immortal Zeus begat—even him whom glorious Hippodameia conceived to Peirithous on ...
“Branch of Ares” is a literal translation of “ὄζος Ἄρηος”, for example in the passage quoted in the question:
οὐκ οἶος, ἅμα τῷ γε Λεοντεὺς ὄζος Ἄρηος
υἱὸς ὑπερθύμοιο Κορώνου Καινεΐδαο:
τοῖς δ᾽ ἅμα τεσσαράκοντα μέλαιναι νῆες ἕποντο.
Homer. The Iliad, book II, lines 745–747. Oxford University Press (1920).
“Branch” here means “descendant”: it is a metaphor ...
Hector is urging his men to attack the Greek ships bravely, but without plundering them afterwards. He says:
If you run away from the battle, or if you lag behind so that you’re not on its front lines, I will make sure you die a dishonorable death anyway. So don’t try to save yourself. Likewise if you stop to plunder the ships.
Edit based on comment from ...