  Teleparallel theories of gravity are described in terms of the tetrad of a
metric and a flat connection with torsion. In this paper, we study spherical
symmetry in a modified teleparallel theory of gravity which is based on an
arbitrary function of the five possible scalars constructed from the
irreducible parts of torsion. This theory is a generalisation of the so-called
New General Relativity theory. We find that only two scalars are different to
zero in spherical symmetry and we solve the corresponding field equations
analytically for conformal Teleparallel gravity, and then perturbatively around
Schwarzschild geometry for the general perturbative theory around GR. Finally,
we compute phenomenological effects from the perturbed solutions such as the
photon sphere, perihelion shift, Shapiro delay, and the light deflection. We
find their correspondent correction to the standard GR contribution and their
dependence on the three model parameters.
