LiPo charging module used as backup?
Microchip's AN1149, "Designing A Li-Ion Battery Charger and Load Sharing System With Microchip’s Stand-Alone Li-Ion Battery Charge Management Controller", describes adding 3 components to a Li-ion charger in order to turn it into a battery backup system.

When battery-powered, the load is supplied through a wide-open PMOSFET. When line-powered, the load is supplied by the line through a single diode drop, and the battery is charged separately. The circuit is designed with the MCP73837 in the middle, but could very easily be adapted for the TP4056.