Just got home from Castle Combe track-day, great day except for a few running issues

Going onto the motorway I gave it some beans and then fuel cut + engine light came on, did my track session trying not to hit over boost again same thing happened

this meant I had to slow down which put me in harms way on very busy session with a wet track, so I was glad when it finished
On the way home I noticed what looked like my petrol was disappearing fast

and the car running ruff so I pulled off the motorway and looked under the car for leaks, , there was none....I then turned off my side lights and Hey Presto the fuel gauge climbed up to almost full again?
So it seems my battery is not holding charge? , would this be the cause of the fuel cut engine light as I had lights on when on track?
A similar thing happened last w/end wile travelling down the M5 with lights on I tried to overtake a lorry and car started throwing a fit an then engine light on
stopped for fuel and the car would not start after spoiling it with optimax petrol

lucky its an easy car to bump by yourself....when I got home turned the engine off then on the off then on..no problem??