how does changing wavelength of radiation on a photoemissive surface change the number of electrons emitted per second?

I was doing an exam question today on photoelectricity, it asked about the effect on electrons emitted per second when the wavelength of incident radiation is decreased. I thought that the number of electrons emitted per second was independent of the energy of each photon, and was rather dependent on the intensity of radiation. However, thats wrong and I just don't understand it. The explanation thats given is something along the lines of, as the wavelength is shorter, the energy of incident photons is greater so less photons are needed to supply the same amount of energy, so more electrons are emitted. I half understand this, but I thought that photons and electrons have 1 to 1 interactions, so why would it matter if a photon can supply more energy to a single electron, wouldn't that just affect the kinetic energy of the electron? I think I'm getting confused with intensity and what that actually means. If anyone could explain how the wavelength of radiation corresponds to number of electrons emitted per second that would be amazing.