We establish that the Hitchin connection is a perturbation of the $L_2$ connection. We notice that such a formulation of the Hitchin connection does not necessarily require the manifold in question possessing a rigid family of Kähler structures. We then proceed to calculate the asymptotic expansion of general perturbations of the $L_2$-connection, and see when under certain assumptions such perturbations are at and projectively at. During the calculations we also found an asymptotic expansion of the projection operator $\pi_{\sigma}^{(k)}$  which projects onto the holomorphic sections of the $k$-th tensor of prequantum line bundle.