Strain- and correlation-tuned magnetic anisotropy and Dzyaloshinskii-Moriya interaction in two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">D</mml:mi> <mml:mrow> <mml:mn>3</m

In the ongoing research for two-dimensional (2D) ferromagnetic materials with strong intrinsic Dzyaloshinskii--Moriya interaction (DMI), most of the efforts are focused on doping, Janus engineering, or heterostructure formation to break inversion symmetry and enhance spin--orbit coupling (SOC). Here, we demonstrate that a pristine 2D material monolayer $H\text{\ensuremath{-}}{\mathrm{FeTe}}_{2}$ can naturally host robust DMI and magnetic anisotropy due to its intrinsic broken inversion symmetry and the strong SOC of Te atoms. We explore the effect of biaxial strain and electron correlation on $H\text{\ensuremath{-}}{\mathrm{FeTe}}_{2}$ using first principles $\mathrm{DFT}+U$ calculations. We systematically investigate the Heisenberg exchange interaction, magnetic anisotropy, and DMI in the space spanned by strain and correlation. Our results reveal a distinct, nonmonotonic strain dependence of both magneto anisotropy energy (MAE) and DMI, including a strain-tunable crossover between in-plane and out-of-plane magnetic easy axes. A remarkable enhancement of the in-plane DMI is observed under the combined influence of strain and strong correlations, which is unusual for pristine 2D materials and suggests a favorable regime for spintronic applications. Notably, even in the absence of strain, $H\text{\ensuremath{-}}{\mathrm{FeTe}}_{2}$ exhibits finite DMI and considerable anisotropy rare for a pure 2D material. Through these findings, we present $H\text{\ensuremath{-}}{\mathrm{FeTe}}_{2}$ as a unique pristine 2D system with robust and tunable spin interactions for exploring fundamental spin orbit-driven magnetic phenomena.