Travelling Problem with Constraints
The problem can be solved efficiently: in $O(K^2 T \log (KT))$ time.

Build a graph where each vertex corresponds to a pair $(t,i)$, where $t$ is a time step and $i$ is one of the nodes that is allowed to be visited at time $t$.  Add edges from $(t,i)$ to $(t+1,j)$ for every $t,i,j$, with a length that corresponds to the travel time from $i$ to $j$.  Add a source node $s$ with edges from $s$ to each $(1,i)$ of length 0, and a sink node $t$ with edges from each $(T,i)$ to $t$ of length 0.  Then, find the shortest path in this graph from $s$ to $T$.

The graph has $O(KT)$ vertices and $O(K^2T)$ edges, so Dijkstra's algorithm for finding shortest paths will run in $O(K^2 T \log (KT))$ time.