solar power ESP8266 microprocessor circuit to turn off when low voltage
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Voltage detection circuit.

I suggest that you need a circuit to hold the reset pin on your micro until solar output is high enough to reliably power the circuit. Figure 1 uses a comparitor to do this function. When the supply voltage divider, R3 - R4, exceeds the R1 - R2 setpoint the micro will be enabled. (You need to figure out setpoints and whether to invert the logic or not.)

Note that you may need to add some hysteresis to this to prevent rapid cycling as you load / unload the solar cell.