Make 9v ups for router with TP4056 and MT3608
Does this circuit will work?


9v*0.5A = 4.5W. Boost converters work on power (with losses), so if 4.5W are going to the router, then at least 4.5W (lets say 5W) must come from the source. 5v/5W = 1A. So 1A must be available from 5V to get 0.5A at 9v.  

USB-C might be able to do this, but to get that much power from a USB-C port, it would have to be negotiated using a special chip. Standard USB ports can only deliver 0.1A. 


  IF USB power is avaliable does MT3608 pull current from higher voltage source (eg: from USB 5v rather then 4.2v from battery)


The boost converter would totally be powered from whichever source was higher (after the voltage drop of the diodes.)

If attempted using a standard USB port, first it will try to run from USB, as that is higher than battery voltage. Exceeding 0.1A will cause the port to shut off, then the battery would power it. So it likely could be charged from the USB port, or operated when unplugged, but not operated while being charged.