Question: I understand when we say current, we mean charge (protons/electrons) passing past a point per second. And the charges have energy due to the e.m.f. of the power supply.

Now tell me, if a lamp has resistance and you hook it in the circuit, how will the current stay the same? The charges obviously lose energy in the lamp and so become SLOWER, which should mean current decreases, right?

[Edit]
All answers explained a bit of everything, so it was hard to choose one. If YOU are looking for an answer, please check the others too, in case the accepted one doesn't answer your question.

Answer: 
The circuit in its whole will settle into an equilibrium state.
If you were to shoot a very short pulse of electrons through your lamp, then indeed, the current would decrease right after the lamp.

But if you have a constant flow, then, as another user illustrated, you'd have electrons bunch up right before the lamp, which limits the output, but ALSO the input! The "traffic jam" works its way backwards, all the way to the source, and eventually slows down all electrons from source to lamp.
@Farcher answer, particularly the last paragraph, sums it up perfectly. The positive work done by the electric field on the charge giving the charge kinetic energy equals the negative work done by the lattice structure that takes away the kinetic energy of the charge increasing the internal energy of the structure. Ultimately, the energy is dissipated as light and heat to the surroundings (a.k.a resistance heating).

A mechanical analog is pushing an object at constant velocity on a surface with friction. The positive work done in pushing the box between two points exactly equals the negative friction work for a net work of zero and no change in the kinetic energy (velocity) of the box. The result is an increase in the temperature at the interface and eventual heat transfer to the surroundings. 

Although not exact, you can think of the external force as analogous to the electric field force, the box analogous to the charge, the velocity analogous to current, and the surface with friction analogous to electrical resistance.

Hope this helps.