Do electrons lose energy between the battery anode and cathode? In a battery circuit, does an electron leaving the anode carry more energy with it than an electron returning, from having been through a machine, to the cathode? If not, how does the battery deliver power to a machine to function by simply transferring electrons from anode to cathode via a circuit?
 A: Electric current is not a flow of energy; it's a flow of charge. Charge and energy are two very different things. 
An electric current is a flowing motion of charged particles, and the *particles do not carry energy* along with them as they move. A current is defined as a flow of charge by I=Q/T; amperes are coulombs of charge flowing per unit time. The term "Electric Current" means the same thing as "charge flow." Electric current is a very slow flow of charges, while energy flows fast. Also, during AC alternating current the charges move slightly back and forth while the energy moves rapidly forward.
Electric energy is quite different than charge. The energy traveling across an electric current is made up of waves in electromagnetic fields and it moves VERY rapidly. Electric energy moves at a completely different speed than electric current, and obviously they are two different things flowing in wires at the same time. Unless we realize that two different things are flowing, we won't understand how circuits work. Indeed, if we believe in a single flowing "electricity," we will have little grasp of basic electrical science.
In an electric circuit, the path of the electric charges is circular, while the path of the energy is not. A battery can send electric energy to a light bulb, and the bulb changes electrical energy into light. The energy does not flow back to the battery again. At the same time, the electric current is different; it is a very slow circular flow, and the electric charges flow through the light bulb filament and all of them flow back out again. They return to the battery.
Electric energy can even flow in a direction opposite to that of the electric current. In a single wire, electric energy can move continuously forward while the direction of the electric current is slowly backwards. In AC circuits the energy flows continuously forward while the charges are alternating back and forth at high frequency. The charges wiggle, while the energy flows forward; electric current is not energy flow.
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William J. Beaty Electricity Misconceptions
A: 
In a battery circuit, does an electron leaving the anode carry more
  energy with it than an electron returning, from having been through a
  machine, to the cathode? If not, how does the battery deliver power to
  a machine to function by simply transferring electrons from anode to
  cathode via a circuit?

Yes, electrons move from the negative terminal of the battery, through the device, to the positive terminal of the battery (in the opposite direction as current flow).  Since they end up at a location of higher electric potential, and they are negatively charged, they have each lost an amount of energy equal to
$$
|q\Delta V|\;,
$$
where q is the electron's charge and $\Delta V$ is the battery voltage.
Inside the battery, chemical energy is used to move more electrons back up to the higher energy region (the negative terminal) where they again "fall" back though the circuit to the positive terminal. Rinse. Repeat. 
