<feel free to move this report from kmenuedit to a more appropriate place, I wasn't sure where to put this>
(this didn't happen until after upgrading from Debian 8.8 to Debian 9.0, or at least I don't remember it happening in Debian 8.x :) )

When kdesudo Kate is run from a shortcut in the tray, the panel will freeze and be unresponsive to mouse clicks until kate is finished.

Note: for these messages I started KDE from an xterm xsession.

From clicking on the tray:
kdeinit5: Got EXT_EXEC '/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu' from launcher.
kdeinit5: preparing to launch '/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu'
QXcbConnection: XCB error: 3 (BadWindow), sequence: 44243, resource id: 125829120, major code: 18 (ChangeProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 44247, resource id: 125829121, major code: 18 (ChangeProperty), minor code: 0
org.kde.kdesu: Daemon not safe (not sgid), not using it.

QXcbConnection: XCB error: 3 (BadWindow), sequence: 44314, resource id: 125829124, major code: 18 (ChangeProperty), minor code: 0

at this point the panel is frozen and not responsive to clicks

If correct password is entered (and then choosing 'Quit' on the 'Session Chooser -- Kate' box):

QXcbConnection: XCB error: 3 (BadWindow), sequence: 60635, resource id: 125829126, major code: 19 (DeleteProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60646, resource id: 125829126, major code: 19 (DeleteProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60647, resource id: 125829126, major code: 18 (ChangeProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60648, resource id: 125829126, major code: 19 (DeleteProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60649, resource id: 125829126, major code: 19 (DeleteProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60650, resource id: 125829126, major code: 19 (DeleteProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60651, resource id: 125829126, major code: 7 (ReparentWindow), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60652, resource id: 125829126, major code: 6 (ChangeSaveSet), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60653, resource id: 125829126, major code: 2 (ChangeWindowAttributes), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60654, resource id: 125829126, major code: 10 (UnmapWindow), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60675, resource id: 125829141, major code: 18 (ChangeProperty), minor code: 0
kdeinit5: PID 21848 terminated.
QXcbConnection: XCB error: 3 (BadWindow), sequence: 60773, resource id: 125829142, major code: 18 (ChangeProperty), minor code: 0
QXcbConnection: XCB error: 3 (BadWindow), sequence: 62253, resource id: 132120590, major code: 18 (ChangeProperty), minor code: 0
kdeinit5: PID 21477 terminated.

then
popup with title "Sorry -- Plasma" saying
KDEInit could not launch '/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu'
Tray is unresponsive to clicks until this box is dealt with

If instead 'Cancel' to password prompt is pressed:

QXcbConnection: XCB error: 3 (BadWindow), sequence: 31048, resource id: 142606340, major code: 18 (ChangeProperty), minor code: 0
Usage: /usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu [options] command
Runs a program with elevated privileges.

Options:
  -c <command>               Specifies the command to run as one string
  -f <file>                  Run command under target uid if <file> is not
                             writable
  -u <user>                  Specifies the target uid
  -n                         Do not keep password
  -s                         Stop the daemon (forgets all passwords)
  -t                         Enable terminal output (no password keeping)
  -p <prio>                  Set priority value: 0 <= prio <= 100, 0 is lowest
  -r                         Use realtime scheduling
  --noignorebutton           Do not display ignore button
  -i <icon name>             Specify icon to use in the password dialog
  -d                         Do not show the command to be run in the dialog
  --attach <winid>           Makes the dialog transient for an X app specified
                             by winid
  --embed <winid>            Embed into a window
  -h, --help                 Displays this help.
  -v, --version              Displays version information.
  --author                   Show author information.
  --license                  Show license information.
  --desktopfile <file name>  The base file name of the desktop entry for this
                             application.

Arguments:
  command                    Specifies the command to run as separate arguments
kdeinit5: PID 23916 terminated.

followed by message box "Sorry --Plasma"
KDEInit could not launch '/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu'

The tray goes back to active but not responding to clicks
After clicking "OK" to this the tray responds to clicks again.

-

Workarounds

add '&' to the command, 
After entering in the password, It'll make the pannel active again, but it'll still be unresponsive to clicks 
After clicking 'OK' to "KDEInit could not launch '/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu'", the tray will be back to normal.
/usr/bin/kdesudo -c "/usr/bin/kate -b %U" &

or just run it from a terminal window
/usr/lib/x86_64-linux-gnu/libexec/kf5/kdesu /usr/bin/kate
org.kde.kdesu: Daemon not safe (not sgid), not using it.
kdesudo kate
QStandardPaths: XDG_RUNTIME_DIR not set, defaulting to '/tmp/user/0/runtime-root'