  We analyze numerically the behaviour of the solutions corresponding to an
Abelian cosmic string taking into account an extension of the Starobinsky
model, where the action of general relativity is replaced by $f(R) = R -
2\Lambda + \eta R^2 + \rho R^m$, with $m > 2$. As an interesting result, we
find that the angular deficit which characterizes the cosmic string decreases
as the parameters $\eta$ and $\rho$ increase. We also find that the cosmic
horizon due to the presence of a cosmological constant is affected in such a
way that it can grows or shrinks, depending on the vacuum expectation value of
the scalar field and on the value of the cosmological constant
