Getting the current path in .desktop EXEC command

According to the Desktop Entry Specification:

Field codes must not be used inside a quoted argument

Consequently, your %k is given literally to the bash command. Changing the Exec line to the following avoids doing so:

Exec=bash -c '"$(dirname "$1")"/run.sh' dummy %k

The above works locally, and also works if there's a space in the path. dummy is given to the bash script as its $0 (what it thinks the script name is), and %k's expansion is available as $1. The nested layers of quoting are necessary to conform to the specification and be space-safe.

Note that %k doesn't necessarily expand to a local file path — it can be a vfolder URI, or empty, and a really portable script should account for those cases too. %k also isn't universally supported itself, so you will need to have some constraints on the systems you expect to use it on anyway.

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As far as debugging goes, you can use ordinary shell redirection under KDE:

Exec=bash -c ... > /tmp/desktop.log

This isn't standardised behaviour, and it doesn't work under GNOME and likely others. You can also give an absolute path to a script crafted to log its arguments and actions in the way you need.