"Draw me a kitten" is reference to any menial task that a supervisor or higher-up assigned for you to perform. Basically something to keep you busy until some "real" work comes along.
"Uggh, Bob's got me drawing kittens all day for him."
(imitating your boss) "Blah, blah, blah, I need you to draw me a kitten, Mike.....and I want it cute! And name it 'Mittens!' And draw a scratching post!"