The photo shows that it can accept 100-240 Volts input which makes it most likely to be a switched mode supply without the mains isolating transformer shown in your schematic. These can be problematic due to some not having the output fully isolated from the input.
Check for any resistance (continuity) reading between the plug pins and the output connector.
Over the gretz bridge rectifier you have "1.2V" dropout, so after it you have "22.8V" plus the transformer is not perfect it might be "23V" not "24V", so "22V" is ok if its an AC to DC adapter. But I dont know if its good for prolong use. Normally a transfomer has to be picked properly, because if it heats up it will break. As for the projects, depends what you are doing. "800mA" can be enough but you have to calculate it for your project.