Being a switching PSU with a fixed output, there is a good probability that the transformer is dimensioned for the minimum voltage required for get the 5V from the regulation circuit without too much unhelpful heat dissipation ..... anyway ..... you can try to check one thing (but BEWARE about the AC section, ok ?)
Cannot be precise from these pics ..... anyway ..... i see that this circuit is a common configuration with an optocoupler (the small black square component with 4 pins), that block the AC oscillator when the output voltage rise over the threshold, and free it when it fall down it ..... this MAY mean (just a possibility) that the transformer can give you more voltage ..... usually, this configuration is controlled from a small 3-pins IC (it looks like a common plastic transistor, but is an IC), that take the output voltage from a dividing resistor network, and trigger the opto when the value go over the threshold level ..... if so, you need to locate the resistor network that take the voltage from the exit, and try to increase a littke bit the value of the "hot side" resistor .....
A LITTLE BIT, ok ? ..... better, changing it with a resistor with a trimmer in serie, and starting to slowly increase it til you see if the voltage reach the level that you need before the PSU goes in unstable mode ot start to do strange noises or "buzz/whistle" noises .....
Just as example, if you find that the resistor network connected to the control IC have a 500 ohm on the "cold" side (from the input pin and the ground), and a 1000 ohm on the "hot" side (from the input pin and the positive), you can try to add, in serie to the 1000 ohm one, a 1000 ohm trimmer, then, powering it with a load (a 12V lamp can be enough) and a multimeter connected , start to rotate it slowly, til the DMM shows the value that you need ..... anyway remember, if in the process, you see that the output becomes "unstable" or the PSU start to make some strange loud, then stop, it means that the PSU cannot reach the voltage that you need, and insisting can burn or blow it .....
If you decide to try this, ALWAYS SAFETY FIRST ..... remember that most of the circuits in the PSU are directly connected to the live AC power, you need to be cautious to not touch anything or short-circuit anything ..... if you have some plastic gloves, better also use them ..... never think "it's just a wall PSU, there's no danger" ..... working with switching PSUs, there's always a danger, if the work is not done in the right way .....
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