Because of the central-limit-theorem, for large $N$ the absolute distance $d_N$ converges in distribution as $$P_N(d_N/\sqrt N)\to p(|X|),$$ where $X$ is a Gaussian random variable with mean zero and variance $2p=1-r$. Since $\mathbb{E}(|X|)=2\sqrt{p/\pi}$ we conclude that $$\mathbb{E}(d_N)\to \sqrt\frac{2(1-r)N}{\pi}.$$

Because of the central-limit-theorem, for large $N$ the absolute distance $d_N$ converges in distribution as $$P_N(d_N/\sqrt N)\to p(|X|),$$ where $X$ is a Gaussian random variable with mean zero and variance $2p=1-r$. Since $\mathbb{E}(|X|)=2\sqrt{p/\pi}$ we conclude that $$\mathbb{E}(d_N)\to \sqrt\frac{2(1-r)N}{\pi}.$$ So this "lazy" random walk differs from the simple random walk by a rescaling of the number of steps by a factor $1-r$.