I'm going to have to give an answer that's very different to Jimmy360's. Apologies.

How does the potential and kinetic energy of a photon relate?

They don't. The photon is all kinetic energy.

Do they mean the same thing?

No. When you drop a brick, its gravitational potential energy is converted into kinetic energy. When you dissipate this kinetic energy as radiation, you're left with a mass deficit, see Wikipedia. From this you know that the potential energy was rest-mass energy. You also know that a photon doesn't have any rest mass, so you ought to know that potential energy doesn't apply.

Also how does De broglie wavelength and potential relate?

An electron has a de Broglie wavelength. When you drop the electron, its gravitational potential energy is converted into kinetic energy, which gets dissipated. The electron then has a mass deficit, and its de Broglie wavelength is increased.

People say that a descending photon is blueshifted, and that it gains energy. But I'm afraid it doesn't. Gravity is not a force in the Newtonian sense. If you send a 511keV photon into a black hole, the black hole mass increases by 511keV/c². Conservation of energy applies. You're like the electron writ large. When you descend you lose energy when you dissipate that kinetic energy. So your total energy is reduced. So you measure the selfsame photon energy as increased. The frequency doesn't actually change, but gravitational time dilation means you and your clocks are going slower, so you measure the frequency as being increased.

I'm going to have to give an answer that's very different to Jimmy360's. Apologies.

How does the potential and kinetic energy of a photon relate?

They don't. The photon is all kinetic energy.

Do they mean the same thing?

No. When you drop a brick, its gravitational potential energy is converted into kinetic energy. When you dissipate this kinetic energy as radiation, you're left with a mass deficit, see Wikipedia. From this you know that the potential energy was rest-mass energy. You also know that a photon doesn't have any rest mass, so you ought to know that potential energy doesn't apply.

Also how does De broglie wavelength and potential relate?

An electron has a de Broglie wavelength. When you drop the electron, its gravitational potential energy is converted into kinetic energy. When you dissipate this kinetic energy as radiation, the electron then has a mass deficit, and its de Broglie wavelength is increased.

People say that a descending photon is blueshifted, and that it gains energy. But I'm afraid it doesn't. Gravity is not a force in the Newtonian sense. If you send a 511keV photon into a black hole, the black hole mass increases by 511keV/c². Conservation of energy applies. You're like the electron writ large. When you descend potential energy is converted into kinetic energy, which gets dissipated. So your total energy is reduced. So you measure the selfsame photon energy as increased. The frequency doesn't actually change, but gravitational time dilation means you and your clocks are going slower, so you measure the frequency as being increased.