As Alan reports, a power bank (phone recharger) may not be best choice for running a low-power Arduino. Power banks are designed to wallop power as fast as possible into a flat smart phone battery, not to trickle milliamps, and their electronics may be unhelpful! Like as not they are designed to stop charging the phone when it’s only taking milliamps,which usually means the phone battery is full.
A plain 6V battery would be my first choice.
On the other hand, Alan is on to something! Power-banks look to the cheapest rechargeable batteries on the market, and they plug straight into any USB device. Good idea then, provided they can be cajoled into trickling!
Dave