If $E = mc^2$ describes the energy-mass relationship of matter, does that mean that if we rearrange to form $m=E/c^2+pc$, does that mean that photons have no energy thus mass = $0$? Or does this mean that photons cannot have both energy and momentum?
This is just a question from a high school partner so no very big words.
Your equation is wrong. The correct form is
$$E^2 = m^2c^4 + p^2c^2$$
Since photons have no mass, it simplifies to
$$E=pc$$
which means that a photon has energy if and only if it has momentum. I don't think a photon with zero energy makes any sense, though. How would you tell it apart from the vacuum?
The $m$ in $E^2 = m^2 c^4 + p^2 c^2$ is the mass of the particle when it is at rest. Photons cannot be at rest and $m=0$ for them; this is what we mean when we say they are "massless". They still have both energy and momentum; if a photon bounces off your head you will recoil just as if it were a little rock (but in practice not by enough to feel anything).
Since no one knows what E^2 means in reality, other than as a mathematical tool, it is interesting that they do not use the photons' kinetic energy in the same manner. Even then, even MC Physics theorizes that photons cannot exist (as we know them) as an elementary particle without velocity. That does not mean that its constituents are massless or have no velocity. Per that theory, photons are formed from mono-charges after their emission from atoms and cease to exist when they are absorbed by other atoms. On atoms (either end of travel) they exist as very active atomic gluons.
Photon mono-charge constituents always have kinetic energy and/ or potential energy.
