In this work, we study the following Kirchhoff type problem $$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$ - ( a + b ∫ Ω | ∇ u | p d x ) Δ p u = g ( x ) u - γ + λ f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where $$p\ge 2$$ p ≥ 2 , $$\Omega $$ Ω is a regular bounded domain in $$\mathbb {R}^N$$ R N , $$(N\ge 3)$$ ( N ≥ 3 ) . Firstly, for $$p>2$$ p > 2 , we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For $$p=2$$ p = 2 , we show, under supplementary condition, the positivity of this solution. Moreover, in the case $$\lambda =0$$ λ = 0 we prove an uniqueness result. We use the variational method to prove our main results.