SUMMARY
Kate snap fails to start unless also the snap kde-frameworks-5 is manually installed in addition. This was tested on a brand new and fully updated installation of Ubuntu 16.04.

STEPS TO REPRODUCE
$ snap version
snap    2.38
snapd   2.38
series  16
ubuntu  16.04
kernel  4.15.0-47-generic

$ snap list
No snaps are installed yet. Try 'snap install hello-world'.

$ sudo snap install kate
kate 19.04.0 from KDE✓ installed

$ snap list
Name                     Version   Rev  Tracking  Publisher   Notes
core18                   20190409  941  stable    canonical✓  base
kate                     19.04.0   26   stable    kde✓        -
kde-frameworks-5-core18  5.56.0    29   stable    kde✓        -

$ snap run kate
qt.qpa.xcb: could not connect to display :0
qt.qpa.plugin: Could not load the Qt platform plugin "xcb" in "" even though it was found.
This application failed to start because no Qt platform plugin could be initialized. Reinstalling the application may fix this problem.

Available platform plugins are: eglfs, linuxfb, minimal, minimalegl, offscreen, vnc, wayland-egl, wayland, wayland-xcomposite-egl, wayland-xcomposite-glx, xcb.

Aborted (core dumped)

$ sudo snap install kde-frameworks-5
Setup snap "core" (6673) security profiles                                                              \2019-04-21T10:28:17+02:00 INFO Waiting for restart...
kde-frameworks-5 5.47.0 from KDE✓ installed

$ sudo snap list
Name                     Version   Rev   Tracking  Publisher   Notes
core                     16-2.38   6673  stable    canonical✓  core
core18                   20190409  941   stable    canonical✓  base
kate                     19.04.0   26    stable    kde✓        -
kde-frameworks-5         5.47.0    27    stable    kde✓        -
kde-frameworks-5-core18  5.56.0    29    stable    kde✓        -

After that, Kate snap runs fine. (except for bug 406396)