The phrase "a small desk on" is correct and usable in written English.
It can be used when describing the location or placement of a small desk in relation to something else.
Example: "There was a small desk on the corner of the room, cluttered with books and papers."
Alternatives: "a small desk at" or "a small desk beside".
Sitting at a small desk on a bare set, Bill Moyers asks scholars, journalists and activists questions about our political system — whom it serves, how it's paid for — that no one else is asking on screen.
To my left, some 50ft from the booths, is a small desk on the edge of the roof, exposed to the elements and with a laptop and a monitor perched precariously on top of it.
Everything Fadi has on display sits on a small desk: a copy of the Koran, in blue leather with ornate gold Arabic script on the cover, and a few boxes of audiotapes that he listens to every day.
The book began on a small desk in the couple's bedroom, where Ms. Hazan wrote out on a legal pad what she could recall of her life.
In the bedroom, a computer sits on a small desk.
The equipment — a computer mounted on a small desk wedged between two file cabinets, in front of a microphone — was ready.
On a small desk by the window lay a magnifying glass and a long stiletto, perhaps the very one she had used for the barrister's murder in "A Certain Justice".
She recommended choosing a small desk that provides enough room to work on a laptop without taking up too much floor space.
Depending on wind speed (measured using a weather station outside the shop), small desk fans automatically turned on and off.
With the right structures in hand, Dr. Watson was one day playing with cardboard cutouts of the four bases when he noticed that an A-T pair on his small desk was identical in shape with a G-C pair.
She rearranged the objects on her small desk — a fountain pen, an inkwell, the collected letters of Hart Crane, two hydrangeas — to make room for her MacBook.
