In this paper, we study the following nonlinear Schrödinger equation with Hardy potential and Sobolev critical exponent \(\begin{aligned} -\Delta u-\frac{\mu }{|x|^2}u+\lambda u=|u|^{2^*-2}u+\beta |u|^{q-2}u \quad \quad \text {in} \ \mathbb {R}^N\backslash \{0\}, \ N\ge 3 \end{aligned}\) having prescribed mass \(\begin{aligned} \int \limits _{\mathbb {R}^N}|u|^2dx=a^2, \end{aligned}\) where \(\beta , a>0\) , \(0 \ne \mu <\overline{\mu }{:}{=}\frac{(N-2)^2}{4}\) , \(q\in (2, 2+\frac{4}{N})\) , \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, and the parameter \(\lambda \in \mathbb {R}\) appears as a Lagrange multiplier. When \(0<\mu <\overline{\mu }\) , combining the Pohožaev manifold, deformation lemma and some analytical skills, we get a normalized solution of mountain-pass type for the problem. Moreover, if \(\mu <0\) , we obtain a ground-state radial normalized solution with negative energy and a mountain-pass-type solution with positive energy. We point out that any nontrivial solution of the problem has the possibility of blow-up at the origin, unlike in the case of \(\mu =0\) , presenting us with a new obstacle. Our results extend the previous one of Li and Zou (Math. Nachr. (2023)).
