Random walk with reflection and skips in linear system

Question

Let's take a case of simple and linear Random walk (0, 1...n) with only one absorbing state n and reflecting state -1, which we can define as:
P (move right at state i) = 1/2 and P (moving left at state i) = 1/2
I know that the expected hitting time at state n, starting at 0 is n² , which can be derived using recursive equation:
E_i = 1/2 + 1/2 * (1 + E_i-1 + E_i)
where E_i is the expected hitting time at state i.
But, what will happen if we change the transition probabilities as follows:
1. At any intermediate state i, walker can jump directly to state n with probability p.
2. The transition probabilities of left and right will be modified to (1-p)/2 each.
This problem is similar to Random re-wiring problem discussed in Collective dynamics of ‘small-world’ net-works by Duncan J. Watts and Steven H. Strogatz .
What is the best way to model this case, and how can we compute the expected time to get absorbed.