We give a new result on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau-Lifshitz-Gilbert model for thermal fluctuations in micromagnetics. We construct strong solutions that belong locally to the spaces $C([s,t);H^1)\cap L^2([s,t);H^2)$, $0\leq s<t\leq T$. It that sense, these maps are a counterpart of the so-called "Struwe solutions" of the deterministic model. We also give a natural criterion of uniqueness that extends A.\ Freire's Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.