  Weak gravitational lensing by black holes and wormholes in the context of
massive gravity (Bebronne and Tinyakov 2009) theory is studied. The particular
solution examined is characterized by two integration constants, the mass $M$
and an extra parameter $S$ namely `scalar charge'. These black hole reduce to
the standard Schwarzschild black hole solutions when the scalar charge is zero
and the mass is positive. In addition, a parameter $\lambda$ in the metric
characterizes so-called 'hair'. The geodesic equations are used to examine the
behavior of the deflection angle in four relevant cases of the parameter
$\lambda$. Then, by introducing a simple coordinate transformation
$r^\lambda=S+v^2$ into the black hole metric, we were able to find a massless
wormhole solution of Einstein-Rosen (ER) \cite{Einstein} type with scalar
charge $S$. The programme is then repeated in terms of the Gauss--Bonnet
theorem in the weak field limit after a method is established to deal with the
angle of deflection using different domains of integration depending on the
parameter $\lambda$. In particular, we have found new analytical results
corresponding to four special cases which generalize the well known deflection
angles reported in the literature. Finally, we have established the time delay
problem in the spacetime of black holes and wormholes, respectively.
