The question is: what is the object in the title called, if it has been defined elsewhere?
Let $G$ be a group generated by a finite set $S\subset G$. A word of length $k$ in $S$ is a string $w=s_1\dots s_k$ where: $\forall i$, $s_i\in S$ or $s_i^{-1}\in S$. Eliminating a subword of the form $s_i^{\mp1}s_{i}^{\pm1}$ is called a reduction, and performing any series of reductions on $w$ until there are no such subwords left results in the unique reduced word $\overline w$.
Now interpret a word $w=s_1\dots s_k$ as a product in $G$, with the empty word corresponding to the identity $e$. The word problem for $G$ is to find an algorithm that determines when $w\sim e$. Every word satisfies $\overline w=e\Rightarrow w\sim e$, but the converse to this only holds when $G$ is the free group over $S$.
One thing we might think about is generalizing the cancellations to subwords $s_i\dots s_{i+j}\sim1$. Applying a sequence of such cancellations does not, in general, determine a unique word like it did with reduction (e.g., trivial subwords can overlap), and the subtleties of this capture important details of the word problem. Motivated by that, the irreducible word problem is to find an algorithm that determines when a word admitting no proper trivial subword is trivial. When $G$ is finite, the length of such words is bounded by $|G|$.
I'm interested in the vocabulary for the mechanics of this, to help me navigate the literature. What do we call the removal of a subword of the form $s_i\dots s_{i+j}\sim1$? (Cancellation?) Finally, what do we call a word having the property that it admits no proper subword equivalent to the identity? (The term "irreducible" would make sense to me if it were consistent with representation theory, but it does not seem to be.) If a geometric approach is desired, these are the 2-cycles and simple closed curves on the Cayley graph.