To make it difficult for someone to win by competing very well; to put up a hard struggle

Everybody knows that the reigning champ is going to win, but his opponent is very tough and I'm sure he will give him a good run

Si sa che il campione in carica vincerà, ma lo sfidante è un osso duro e sono sicuro che gli darà del filo da torcere

The Democrat won the election by a very narrow margin, the Republican candidate having given him a good run for his money

Il candidato Democratico ha vinto il ballottaggio ma solo per pochissimi voti, dato che il rappresentante Repubblicano gli ha dato molto filo da torcere

If you are betting on a horse and it doesn't win but comes close, you have lost your money, but at least had a run for it