These expressions have very similar meanings and they are both used in informal contexts. Let’s start with the first one.
To see a man about a dog is to go to the bathroom. It is an indirect (and hence more polite) way of saying that you wish to use the toilet.
- Make a strong tea for me. In the meanwhile, I have to see a man about a dog.
- Excuse me. I am going to see a man about a dog. I will be back in a minute.
This expression is also used in those situations when you want to keep your destination or the purpose of your visit a secret.
- I don’t know where Rahul has gone. He just said that he had to see a man about a dog. (Rahul doesn’t reveal his destination. He just says that he has to go.)
- Rohan is very secretive. He never says anything about his frequent trips. If you ask, he will just say that he has to see a man about a dog.
Women often use the phrase powder my nose to express their desire to use the bathroom. This phrase is not normally used by men.
- Excuse me. I wish to powder my nose.