AC powered microcontroller issue
It looks like a relatively standard capacitive dropper circuit.  C1 provides the current limiting. R1 prevents a surge on power-on.  The two diodes, plus the two pairs of diodes (why pairs?) form a bridge rectifier, with C2 providing the smoothing.

With the correct load applied, there should be an appropriate voltage on the + and - terminals.  Without the load, the voltage could rise to the point where C2 goes bang.

You should be able to test each of the components in turn with a multimeter.  If C1 is failing, it may not be letting through enough current.

Beware, there would normally be a high value resistor across C1 in similar circuits.  Without that resistor, if the circuit has been plugged in recently, there could be a substantial voltage across C1.