The phrase "at desk" is correct and usable in written English.
You can use it when indicating the location of an object or person. For example: "I left my book at desk this morning."
Worked at desk in offices of Bond and McEnany.
Brodsky, J. [Photograph of Joseph Brodsky at desk with typewriter].
8 30 At desk, which is piled with file folders.
Announced himself, but elderly "boy" at desk insisted he fill out customary blank.
Threw no cans in bucket, just sat very still at desk.
Pizzella, a soft-spoken, mustachioed man with the swept-back hair and insouciant good looks of a forties movie villain, works as an at-desk masseur specializing in high-tech companies.
As well as the regular phone calls, PRs arrange at-desk visits, parties and press releases.
(If anecdotal evidence is any indication, some workers even feel awkward about accepting the brief at-desk massages often offered as corporate perks).
Kafka's "Castle" is explicitly invoked; and so, implicitly by the unfinished clerk-at-desk play, is the entirety of Beckett's drama.
Pieces of such an approach show up in the bill-status attribute used by the House for its XML schemas and DTDs, which contains such status indicators as "held-at-desk-House" and "reengrossed-amendment-Senate".
