Try At -> Function[{t, x, y, z}, t^2 + x^3 + y^4 + z^5]. The symbol At appears as a function, not a formula.

For posting the code, try Edit > Copy As > Input Text or converting to InputForm. The boxes are rather hard to read. It also should be formatted as code in MarkDown, or it won't copy reliably.

You're welcome! BTW, indicating partial derivatives with a vector of nonnegative integers (sometimes called a "multi-index") is a standard notation in higher mathematical analysis, often first encountered in grad school. So it's standard but not commonly known. Your implicit suggestion that it be mentioned in the documentation seems a good one, perhaps on the pages for D and for Derivative. The "Details" section often mention how certain symbols are formatted. Another tip: You can use FullForm to see that the partial derivative is in fact Derivative[1, 0, 0, 0][At][t, x, y, z], but the Front End displays in the standard, if arcane, notation.

I know you weren't asking for the following, but it writes the derivatives of At in the way you'd like, I think:

MakeBoxes[Derivative[i_, j_, k_, l_][At][tt_, xx_, yy_, zz_], 
  TraditionalForm] := Block[{t, x, y, z},
  MakeBoxes[D[At[tt, xx, yy, zz], ##], TraditionalForm] & @@ 
   DeleteCases[{{t, i}, {x, j}, {y, k}, {x, l}} /. {v_, 1} :> v, {_, 0}]
  ];

D[At[t, x, y, z], t, t, x] // TraditionalForm
(* shows the traditional Leibniz form of the partial derivative *)

Of course it works only for At. There are issues implementing it in general. Like determining the symbol names for the arguments of every function that might be differentiated at some time by some user.