Custom taxonomy admin description

I don’t know of any hooks to add or filter the metabox so you can either create your own metabox to replace the custom taxonomy metabox or in this case just use JQuery .append()

for example is your taxonomy is named banners and its none – hierarchical (like tags) then:

$('.tagsdiv-banners').append('<p>your short description here</p>');

and if its hierarchical (like categories)

 $('.bannersdiv').append('<p>your short description here</p>');

the only difference is the name of the selector , for none hierarchical taxonomies you use tagsdiv-{$tax_name} and for hierarchical taxonomies you use {$tax_name}div.