Circuit to maintain a minimum current?
According to the data sheet: -


  The TP4056 automatically terminates the charge cycle when the charge
  current drops to 1/10th the programmed value after the final float
  voltage is reached.


And there is a resistor (\$R_{PROG}\$) which can be set to something like 10k limiting the output current to 130mA or it can be set to 5k for 250mA but only you know where to set it to for normal operation.

If your running current is less than say 250mA and it doesn't drop to less than 25mA this is do-able with a 5k resistor. The trick is not exceeding the 10:1 difference in currents taken by your circuit.

Here is the link to the document - take note of the table at the top of page 3.