Recent Post by Category ID

Hello,
I’m writing my first wp plugin, for picking up recent post from selected categories, putting ids, separeted by commas, in an input text form.
I’ve followed a tutorial and changed something, the plugin work, but only if I write manually cat ids in the code. The input in widget don’t works.
I post the code.

[Code moderated as per the Forum Rules. Please use the pastebin]

Tank u all,
Ricky