Using your mobile device with an external battery connected to it all the time creates a lot of conflicting situations. There's a big difference in how your device is working when running off its internal battery vs. when you have an external battery connected to it. The former is a typical state, the latter is in charging mode. Here's some good reading on what's going on:
https://batteryuniversity.com/learn/article/charging_lithium_ion_batteries
also, since that external battery will be using your USB port (microUSB or Type-C):
https://batteryuniversity.com/learn/article/charging_from_a_usb_port
Not knowing just which model you have or any other relevant details to your question, it's hard to be more definitive but generally if the battery in your device has failed and it has a user-replaceable one you'd be better off just replacing the internal battery than using an external one. Again, depending on what model a non user-replaceable battery 'might' still be readily replaceable, but that's very, very dependent on what model as some are relatively easy to get to, some require a complete tear-down and are secured in place with lots of glues and adhesives.
But if replacing the battery isn't an option, and you do have to rely on an external battery as a stop gap measure, there are some phone cases that have integral batteries. But at that point, instead of buying one of those you might want to devote those funds to just upgrading to a new(er) phone. Just weigh your options.