Let \(k\) be an algebraically closed field of characteristic zero and let \(k(t)\) be the field of rational functions over \(k\). Further, let \(\mathbb{K}\) be a finite extension of \(k(t)\). For given non-zero elements \(f_ 1,\ldots,f_ n,g\) of \(\mathbb{K}[X_ 1,\ldots,X_ n]\) \((n\geq 2)\), consider the equation \[ \sum^ n_{i=1}f_ i({\mathbf x})\cdot x_ i^{r_ i}=g({\mathbf x}) \tag{*} \] in \({\mathbf x}=(x_ 1,\ldots,x_ n)\in\mathbb{K}^ n\) and \({\mathbf r}=(r_ 1,\ldots,r_ n)\in\mathbb{Z}^ n\) under the condition that the sum on the left hand side of \((*)\) has no proper vanishing subsums. The main theorem of this paper gives an effective upper bound for the height (in its usual sense) of the solutions to \((*)\). Applications are given to a few special equations.