Power drop on AC-DC converter due to relay
Your last comment is confusing/incorrect.  The power adapter current rating is how much current ( amps ) the supply can provide.


  This happens with a cheap AC-DC converter we bought on Aliexpress but not with other more expensive converters like VSK-S1 


I think you have your answer.  Those cheap adapters from China are not always what they claim!  Here is an EDN article, showing the shoddy and dangerous things coming from China.

Teardown--12V-AC-adapters---The-Horror

There are several blogs showing cheap-o supplies rated for 1Amp can only provide 0.5Amps!  

Also, there is more to power supply hold up than just more caps.  The supply has a transient response based upon its control loop.  If that is not well designed, then adding bulk caps is just a band-aid.
Stick with a quality supply and your woes will be far less, and your product will work better and last longer.