It deals with the Battery and its condition: Meaning, It's Ability to Charge, hold that Charge, and deliver this Charge at a given time, rate and with little voltage drop.
The Air Conditioning system is very heavy current dependent and also places a load on the engine while it's in use.
So if that say you have a weak battery - change it out - have them test it first though, for a good tester can be used to check the charging system - may indicate other things like corrosion which affects the contacts for the wires - the charging system uses cables, when they're bad or corroded - they can't deliver power when and where it's needed if they don't have a good path for current flow thru to get power where it's needed..
To have the A/C come back on after some driving - means the battery was attempting to recharge as the A/C worked.
IF this continues, have them inspect, repair and/or possibly re-charge the A/C system too...
Using A/C; it drained down the battery faster than it can keep up the voltage and power the A/C - and the rest of the system supplied by the Alternator - to keep up the demand - so the car sees this and it sheds loads, in this case. The AC unit stopped working until the battery recovered.
Have the service shop inspect and test the charging system, and it's wiring to the Battery. Test it - don't just change out the battery - there's more to this than a simple weak battery diagnosis. They should test the system and fix all of the problem, not just get you a new battery. You might be back in that shop in less time than this first time you found this out.
AGM? These are not designed for AGM unless it's been modified to accept AGM-based battery in a standard car charging system - Which nearly all Fiesta's uses a regulation system that monitors the Alternators output thru a sense wire and compares it to the battery - AGM works a little differently and most analog-based Alternator sense regulation systems are not programmed, designed or have the technology to deliver the power profile the AGM battery would need and still supply power to the vehicle without frying its electronics in the process.
Why? Mostly due to the construction differences between lead-acid and AGM. Correct, - for Pb is used in both types, yes, but the material used to separate the plates is what gives them their characteristics that most older Charging systems can't tolerate.
When a wet (Pb-plate flooded) battery charges it looks like nearly a dead short for a time as the alternator then tries to boost power into the wires to feed the system - which the battery as you know is connected to also. So the Alternator just dumps power into the wires to let it go to where it needs it the most but it controls it using a method to sense the load by observing how the voltage range changes when it changes current flow - as an analog feedback principle of negative feedback loop.
In AGM the charge rate is much quicker and can force an older alternator to crowbar (shut-down) and make it think it's over-regulating so it drops power but it will spike upwards again as the AGM demands and absorbs this charge at a much quicker rate than the typical older Pb battery - this puts the entire charging system at risk for spikes which can damage the older electronics not built to help regulate power thru their own independent internal-self regulation which the newer technology designs use which lessens the impact from those spikes that can damage their otherwise unprotected systems.
The Start-Stop engine systems use AGM to re-charge faster in less time than the older Pb ones - which again is why the older analog-based regulation charge sense systems are not able to properly utilize an AGM battery - best to stick with an older Pb design.
