What's the word or phrase for the situation in which you know about someone's secret and that gives you an advantage over them? [closed]

From comments under the question, this is asking for a term to describe the knowledge of the secret itself, and how it can be used. (In short, a better word than just advantage.)

To have knowledge of a secret that could be used against somebody is to have leverage over them:

[Merriam-Webster]
2 : POWER, EFFECTIVENESS
// trying to gain more political leverage
// The union's size gave it leverage in the labor contract negotiations.
// The player's popularity has given him a great deal of leverage with the owners of the team.

In the example in the question:

Knowing an embarrassing secret, a child has leverage over his father.

In a more informal sense, it could also be said that you have dirt on them:

[Merriam-Webster]
3 e : embarrassing or incriminating information
// trying to dig up dirt on her political rivals

Of course, the two words are not mutually exclusive:

Having dug up dirt on him, she had leverage over him.

Once all of this is in place and actually used, it would then be described as blackmail, extortion, or just coercion, depending on the context.