Macdonald's ninth variation of Schur functions is a broad generalization of the classical Schur function and its variants, defined via the Jacobi-Trudi determinant formula. In this paper, we establish various algebraic relations for $S^{(r)}_{λ/μ}(X)$, a class of the ninth variation introduced by Nakagawa, Noumi, Shirakawa, and Yamada, by combining the Jacobi-Trudi formula with determinant formulas such as the Desnanot-Jacobi adjoint matrix theorem and the Plücker relations, which generalize the corresponding relations for Schur functions. As an application, we investigate algebraic relations for "diagonally constant" Schur multiple zeta functions and examine their specific special values when the shape is rectangular.