Surgery here is turned on its head. Instead of the surgeon piercing a patient, the speaker imagines piercing the surgeon, right into his "living clock." The phrase suggests that even the person who holds life in his hands is a ticking, mortal thing, with a body that keeps time like any other.
Inside that clock, the speaker finds "fine times / better than Wednesday next." That's a quietly funny line. Wednesday next is an appointment, a slot in a diary, the ordinary scheduled time of a procedure. The poem hints that there are better moments to be found than the ones medicine sets out for us. Then the tone cools: those moments are "to be dispatched / with a little wipe," as if a life's worth of time could be cleaned away like a smear on an instrument.
The mood is dry, clipped and a little unsettling. The speaker seems wry rather than angry, amused and uneasy at how casually mortality gets handled.