  Motivated by the recent work on $T\bar{T}$-type deformations of 2D CFTs, a
especial class of single-trace deformations of AdS$_3$/CFT$_2$ correspondence
has been investigated. From the worldsheet perspective, this corresponds to a
marginal deformation of the $\sigma $-model on AdS$_3$ that yields a string
background that interpolates between AdS$_3$ and a flat linear dilaton
solution. Here, with the intention of studying this worldsheet CFT further, we
consider it in the presence of a boundary. In a previous paper, we computed
different correlation functions of this theory on the disk, including the bulk
1-point function, the boundary-boundary 2-point function, and the bulk-boundary
2-point function. This led us to compute the anomalous dimension of both bulk
and boundary vertex operators, which first required a proper regularization of
the ultraviolet divergences of the conformal integrals. Here, we extend the
analysis by computing the bulk-bulk 2-point function on the disk and other
observables on the sphere. We prove that the renormalization of the vertex
operators proposed in our previous works is consistent with the form of the
sphere $N$-point functions.
