I'm not sure that I understand why the other answers suggest that it's not a common word or that 'tug at' is the best/correct option. I wonder if this is another example of a difference between BrEng and AmEng, because it seems to me that both are perfectly normal and common in Britain. But I couldn't find any comments about this in a dictionary, so maybe it's just a bit less fashionable in N America and other parts of the world? There is possibly a slight sense of difference between 'to tug a rope' and 'to tug at a rope', because it's possible to argue that if you tug a rope something (on the other end of the rope) moves to a new position and stays in that new position until it is moved again... whereas if you tug at a rope, the rope (and the thing on the end of it) either doesn't move or returns to the original position almost immediately. So for example: "Two teams in a tug-of-war competition tug a rope, and win by pulling the other team across the line." OR "The bell ringers tugged at the bell ropes to celebrate the wedding." But I think there's probably a lot of confusion in everyday use and you'll find examples where "to tug" and "to tug at" have exactly the same meaning.

Examples. (It’s not a very common word. ) Our boat was out of fuel. Another boat tugged us to the dock. (You wouldn’t normally tug the rope, you’d tug something else with a rope) We got a tug to the dock. Stop tugging my hair! (Pulling. Has a negative connotation) I tugged on the rope. It didn’t budge. (Pulled hard) I tugged at the rope. (Pulled, but with less force, less purpose) In PE, we played tug of war today.

Both sentences are okay to say. However, the second sentence is better/more formal.

Hello! It's the second sentence that is correct! :)