We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators and prove this conjecture in several cases. Finally, applications of these results are given to left n-invertible elementary operators and essentially left n-invertible operators.