A lavage unit (50) includes a handpiece housing a pump (560, 620) to which an irrigation tube (550) is connected. The pump includes a bellows (560, 620) defining a pump chamber (576, 624). When the bellows is reciprocated by a linkage (602, 608), the bellows draws irrigation fluid from the irrigation tube (540) into the pump chamber and then discharges the fluid out of the pump chamber through an outlet (590). The pump further includes a section (566, 628) formed from compressible or flexible material that defines a supplemental chamber (568, 630). The linkage (602, 608) is connected to said pump to alternatively compress the bellows forming the pump chamber (576, 624) and the material forming the pump supplemental chamber (568, 630). During the movement of the linkage a valve (582, 638) located between the pump chamber and the supplemental chamber reciprocates with the linkage.