  Determining the index of the Simon congruence is a long outstanding open
problem. Two words $u$ and $v$ are called Simon congruent if they have the same
set of scattered factors, which are parts of the word in the correct order but
not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of
$\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality,
we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered
factors of length $k$ are absent, w.r.t. Simon congruence. We present a full
characterisation as well as the index of the congruence for $m=1$. For $m\neq
1$, we show some results if in addition $w$ is $(k-1)$-universal as well as
some further insights for different $m$.
