give someone the elbow
give (someone) the elbow
1. To spurn, jilt, or reject someone, especially a lover or romantic interest. Jonathan has had a broken heart ever since David gave him the elbow.
2. To summarily dismiss or oust someone from employment; to fire someone. Management promptly gave the new accountant the elbow after his miscalculation cost the company hundreds of thousands of dollars. They'll give me the elbow if they ever find out I came into work drunk yesterday.
give someone the elbowreject or dismiss someone. informal
The image is of nudging someone aside in a rough or contemptuous manner.