get the idea, that a photon is some kind of physical object, out of your mind. light is energy, that is transmitted over the electromagnetic field as a wave. but that wave can not take arbitrary values but only discreet multiples of a fixed value. kind of a packet of energy. and we call this packet of energy a photon. light is still an electromagnetic wave but it can only be produced and absorbed photon per photon.

there is this electromagnetic wave. and if this wave has the right wavelength (=energy density), it can excite a special molecule (the chlorophyll) in a special way that it can break the chemimal bonds between carbon dioxide molecules (co2) and water molecules (h2o) to form glucose molecules (c6h12o6) releasing some free oxygen in the process. (of course it is much more complicated than that). the chemical energy stored in glucose is higher than that of carbon dioxide or water. thats the reason, why the plant does this in the first place - glucose can be used to store energy. that energy must come from somewhere. and you guessed it: it is transformed from the light, the chlorophyll has absorbed.

the speed of the wave is irrelevant. one moment, there is no wave (just the "flat" electromagnetic field) and the chlorophyll is kind of in a dormant state. the next moment, there is a wave, consisting of - lets say, 10 packets of energy (10 photons). that energy is enough to excite the chlorophyll. it can catalyse carbon dioxide and water to glucose and free oxygen. the next moment, there is no wave anymore. aka: no more energy packets. the chlorophyll gets dormant again. photosynthesis stops.

it's not miniature balls zipping past at incredible speed! it's energy transmitted as waves. the speed of that wave is irrelevant. what matters is: is there NOW enough energy you can sap from the electromagnetic field to do something or not.

and to hammer down the obvious: if you turn on the light for 10 seconds in an otherwise dark room, there is kind of a wave front, that lasts for 10 seconds. in that 10 seconds, the electromagnetic field in direct vicinity around the chlorophyll molecule carries enough energy to do something. you only notice, that this wave arrived at the speed of light, if you compare the field strenghs of your own point with other points in the distance.