Hydrogels are of interest for a wide range of applications. The ability to control when the hydrogel degrades can provide beneficial properties such as controlled degradation in the environment or the stimulated release of drugs or cells. Self‐immolative polymers are a class of degradable polymers that undergo complete end‐to‐end depolymerization upon the application of a stimulus. They have been explored for hydrogel development, but the ability to prepare and selectively degrade self‐immolative hydrogels under neutral aqueous conditions has so far been limited. We describe here the preparation of water‐soluble polyglyoxyamides with cross‐linkable pendent azides and their cross‐linking to form hydrogels with 4‐arm poly(ethylene glycol)s having unstrained and strained alkynes using copper‐assisted and strain‐promoted azide‐alkyne click chemistry respectively. The influence of pendent azide density and solution polymer content on the resulting hydrogels was evaluated. A polyglyoxylamide with a 70:30 ratio of pendent hydroxyl:azide successfully provided hydrogels with compressive moduli ranging from 1.3 – 6.3 kPa under copper‐free conditions at 10 – 20% (w/w) of polymer in phosphate‐buffered saline. Selective depolymerization and degradation of the hydrogels upon irradiation with light was demonstrated, resulting in reductions in the compressive moduli and the release of depolymerization products that were detected by NMR spectroscopy.