Chinese power supply schematics/components
To give a complete correct answer to this is difficult. The best you can do is look for the UC2384AN application note and find out and understand how such a system is configured. Then try also to unsolder the burned parts carefully and measure them if possible. Most probably the component (transistor) under te heatsink has a short circuit.
Next you have BD1 four diodes in a bridge, This is the incomming rectifier after the filter. After taking out the switching transistor and verifying the big capacitors (they seem to bulge and need replacement. Any bulging capacitor need to repaced anyhow) you should be able to connect the mains. (Carefull for you health) You should then measure mains voltage *1.4. over the capacitors.
Further help is difficult without a schematic. So I suggest you make one and mention the parts and values found and go from there. If things become really difficult show the schematic and we might be able to help more. 

Here is the link: http://www.ti.com/lit/an/slua143/slua143.pdf