The main result of the paper is a canonical description of free objects in the variety $\mathcal U(M;N)$ of groupoids with the following axioms: \[ \{x^{m+1}\cdot y=xy\mid mı M\}\cup\{x\cdot y^{n+1}=xy\mid nı N\} \] where $M$ and $N$ are sets of positive integers, such that $M\cup N\neq\emptyset$. Applying the obtained description, corresponding characterization of free subgroupoids of a $\mathcal U(M;N)$-free groupoid is given.