Can I hook up a single phase switch to an electric wood splitter?
"THIS MACHINE KILLS"

WHAT YOU ARE TRYING TO DO HAS "ISSUES" EVEN WHEN DONE "PROPERLY.

Here I've copied material from two comments by J:


From the manual : For your safety, your log splitter is equipped with a “ZHB” control system that requires both hands of the operator. The left hand controls the hydraulic control lever while the right hand controls the pushbutton switch. The log splitter will stop if the lever or pushbutton are released. Only after the lever and pushbutton are released will the log ram return to its starting position.
It's worth noting that either OP's splitter has a different HY56, OR the HY56 is installed with the red lever permanently depressed (ie: the green button is a momentary switch, not latching). This switch is a safety switch whose only purpose is to force your hand to not be between the log and the splitting blade while you're working the hydraulic lever. 




The HY56 


Is a double pole single throw switch - it has a contact in both mains leads so the equipment is fully isolated when the switch is off.
Latches on when the green button is pressed and "trips" off when the red button or associated larger red lever are pressed.  


It is not safely sensible to wire a single pole external switch to it as 


You only switch one of the two poles.
This either


will not turn the equipment on or
needs one of the two poles in the HY56 to be bridged.
This would be stupid lethal and probably illegal.



You could use a two pole 4 wire foot switch with each external switch bridging one of the internal poles.  This has potential safety issues as the internal and external switches can turn the equipment on. Depending on circumstance it may not be possible to turn the equipment off in an emergency. eg if foot switch is latching or if non latching and a log or user falls on it and holds it activated it the internal safety switch would not function.
If turned on with the internal switch and the external switch is held activated the internal cutout will work as intended but the equipment will not stop.

While confused internal / external interaction may only take a second or so to sort out, lives and limbs can be lost sooner than that.

The switch paths cross the switch "sideways" - not end to end. There is enough on the web re how the switch is wired that I am NOT going to expand on that or provide a diagram. If you cannot work out and implement with certainty a failsafe way of doing things from what I have written then you should be seeking help 'in the flesh' from an accomplished practitioner.

A second external HY56 with an external/internal changeover switch would deal with some but not all of the above issues.

A second external HY56 wired in parallel with the existing switch would be a mind-confusing man-trap waiting to maim or kill you.