The act of cowardice that Godrich performed was leaving the battlefield. When Lord Byrthnoth was killed, all his retainers were bound by oath to stay in the fight and avenge his death. Some of them, courageously stayed. Others cowardly fled. Godrich was one of the latter.
The poet finds this especially cowardly because not only was he one of the first to escape, along with his brothers, but also because he fled in his lord's own horse. (The author specifies "He leapt upon the very stead his lord had owned, his saddle, which was not right at all".)