Need advice re battery-powered Pi2

Hello! I’m trying to make a portable Rpi2-based computer, powered from batteries. After a brief experiment with my old RPiB and a rechargable phone charger (which seemed to work quite well) , I innocently thought that that was problem solved (by the way, whilst I’m an experienced computer user and can program a bit, I’ve extremely little knowledge of or experience of playing with electronics). The screen I’m using is the 5in non-touch one that’s recently been released, incidentally.

I bought a second phone charger (the large ones of the Kit brand, with both 1A and 2.1A power outputs) so that the screen would have its own power supply. Initially, I tried running both off the 1A socket, but this seemed a bit iffy, and I noticed occasionally I was getting a wee path of varicoloured squares at top right of the screen, but it was running. I then tried to read the contents of a USB stick. This caused the setup to switch off every timne. Eventyually I realised that there wasn;t enough power to handle the USB stick, did a quick bit of research and found that it should be OK to power the RPi from the 2A supply. This did indeed make things better, but not only was teh USB stick still not read, the whole setup is still switching itself off after only about 20 seconds after the desktop appears.

If the RPi is powered from a mains USB power lump, it behaves normally, as one would expect, and doesn’t power off. Oh, and I;ve also discovered that the cable that comes with the phone chargers don;t work as well as another with the same end connectors that I had lying around. If some kind person could explain why the RPi tends to switch off after less than a minute of powered from the rechargable battery (even though there’s plenty of juice in it) and what I need to do differently, 'twould be greatly appreciated. Oh, and is it possible to get an on-off switch to control the power rather than physically plugging/pulling the cable out?