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 question 

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> The charges obviously lose energy in the lamp and so become SLOWER, which should mean current decreases, right? 
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shows that you have a misconception about the motion of the conduction electrons. 

If you were correct then to maintain the same current around a circuit by a miracle more conduction electrons would need to contribute to the conduction process as you went around the circuit.  

If this did not happen then the conduction electrons would move slower and slower and . . . . . eventually stop? 

In fact what happens is that the electrons gain kinetic energy (and lose electric potential energy) between collision with the lattice (bound) ions from the electric field in the wire and then lose that extra kinetic energy to the lattice ions upon collision with them.  

The net effect is that the temperature of the material increases as the internal kinetic energy of the material has increased (the lattice ions vibrate more) and the conduction electrons move along the conductor with a constant average speed.
@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.