Which Specialized pump do you have? Are they easily rebuildable? Thanks

I don't own one a Specialized but they sell the rebuild kit, it can't be that difficult they're all pretty much the same, usually you just unscrew the top where the plunger goes into the body of the pump, then the plunger comes right out. Some of the pumps have set screw at the top on the body cap where the plunger goes into, remove that set screw, and then either it comes out with a back and forth twisting action or you then have to unscrew the top part, it behooves you not to loose the little screw! Then before you replace the seal, wrap a thin cloth around it and the seal head and use the plunger as a cleaning thing by running the cloth up and down in the tube, if you still see gunk inside then you'll need to soak that rag in Acetone and repeat the cleaning process. Next take some steel wool to the plunger rod and get any grime or corrosion off of it. When done with the cleaning take the rag off and pry off the seal and grease the all around the area that the seal was on, you pretty much want to cover the head that the seal goes on with a nice thick layer of grease, then install the new seal, reassemble the plunger and now it should work like new.

See this video for a Lezyne: