I have also occassionally noticed fluorescence quenching following EdU click reactions. I would consider the following:

1) Ensure that the effect you are seeing is not caused by fixative procedures (i.e., formalin/formaldehyde is known to cause signal decrease)

2) ROS are formed during the reaction; try the reaction with different amounts of ascorbic acid (or whatever the supplier names the last compound you add to the click mix). You can add half, or double the recommended amount and still have a function click-reaction (with varying efficiency of course). Then check whether you have dose-dependent quenching of your GFP signal

3) Consider the use of tris(hydroxypropyltriazolyl) methylamine (THPTA) to reduce, although I haven't had success with this compound yet.

This paper touches upon some of these concepts

https://doi.org/10.1002/biot.201400026

Best,

Robin