  The notion of Turing kernelization investigates whether a polynomial-time
algorithm can solve an NP-hard problem, when it is aided by an oracle that can
be queried for the answers to bounded-size subproblems. One of the main open
problems in this direction is whether k-Path admits a polynomial Turing kernel:
can a polynomial-time algorithm determine whether an undirected graph has a
simple path of length k, using an oracle that answers queries of size poly(k)?
  We show this can be done when the input graph avoids a fixed graph H as a
topological minor, thereby significantly generalizing an earlier result for
bounded-degree and $K_{3,t}$-minor-free graphs. Moreover, we show that k-Path
even admits a polynomial Turing kernel when the input graph is not
H-topological-minor-free itself, but contains a known vertex modulator of size
bounded polynomially in the parameter, whose deletion makes it so. To obtain
our results, we build on the graph minors decomposition to show that any
H-topological-minor-free graph that does not contain a k-path, has a separation
that can safely be reduced after communication with the oracle.
