Would Hank have been able to let Walter Jr. off for attempting to buy alcohol so easily?

My interpretation of this scene is not that Hank has the power to do this, but that he is able to talk "one law enforcement officer to another". Hank is in the DEA, not the local police force, so although undoubtedly he would be considered more senior, I don't think he would be able to 'pull rank' to get Walt Jr out of trouble.

I don't believe that the cop was undercover, I think he was off-duty and happened to just be going into the gas station when he was interrupted by Walt Jr.

The cop probably wants to go back to enjoying his evening ... not dealing with a foolish kid buying alcohol. Buying alcohol isn't exactly the most heinous crime. I can imagine Hank promising to deal with him, and asking the cop to cut him some slack.

The fact that the cop could hand Walt Jr over to his responsible law enforcement "father" probably made the cop's life a lot simpler and he could go back to what he was doing.

I disagree with the accepted answer — I don’t think Hank ever mentioned his job or pulled a favor with the off-duty cop. Remember that the cop tells Junior, “You're lucky you've got a good dad here”, meaning Hank, so I’d be surprised if Hank was even properly introduced to him, let alone announced his role as a federal law enforcement agent. This also clearly shows that Hank didn’t know this police officer, making it harder to get a favor from him.

I read the situation as the cop threatening to arrest Junior and making him call a parent as a way of scaring him away from criminal behavior. Since Junior was angry with his parents over Walt’s cancer diagnosis (and possibly just to avoid getting into trouble at home), he called Hank instead and introduced him as his father. Hank, upset at usurping Walt’s rightful role, quickly made his apologies and promised adequate punishment, and the cop let Junior go with a warning, impressed with the speed with which Hank responded and the care he showed towards Junior.

Walt Jr. lied. Hank openly supported the lie. We don't know whether

• Walt Jr. managed to call Hank in advance to arrange it, or
• lied to the police claiming Hank was his father, and they called Hank and said "we have your son Walt" and Hank realized the game and decided to go along with it instead of bust it.

We don't know. If the latter case, Walt Jr. took a huge, huge gamble and it paid off... not knowing which makes it interesting.

Regardless, the police were led to believe Walt Jr. was Hank's son, not Walter's.

The cops agreeing to drop it is a professional courtesy, cop to cop. It is not a "get out of jail free" card. They reasonably expect the father to handle the discipline problem "domestically", and not see that child in their station again. This is reasonable practice, because they know what kind of person a cop is.

Breaking Bad didn't spend a lot of time with it, because it's a trope that most TV watchers will be familiar with.