  Let $n,m\in \mathbb{N}$, and let $B_{n,m}(\mathbb{R}P^2)$ be the set of $(n +
m)$-braids of the projective plane whose associated permutation lies in the
subgroup $S_n\times S_m$ of the symmetric group $S_{n+m}$.
  We study the splitting problem of the following generalisation of the
Fadell-Neuwirth short exact sequence:
  $$1\rightarrow B_m(\mathbb{R}P^2 \setminus \{x_1,\dots,x_n\})\rightarrow
  B_{n,m}(\mathbb{R}P^2)\xrightarrow{\bar{q}} B_n(\mathbb{R}P^2)\rightarrow
1,$$ where the map $\bar{q}$ can be considered geometrically as the epimorphism
that forgets the last $m$ strands, as well as the existence of a section of the
corresponding fibration $q:F_{n+m}(\mathbb{R}P^2)/S_n\times S_m\to
F_{n}(\mathbb{R}P^2)/S_n$, where we denote by $F_n(\mathbb{R}P^2)$ the $n^{th}$
ordered configuration space of the projective plane $\mathbb{R}P^2$.
  Our main results are the following: if $n=1$ the homomorphism $\bar{q}$ and
the corresponding fibration $q$ admits no section, while if $n=2$, then
$\bar{q}$ and $q$ admit a section. For $n\geq 3$, we show that if $\bar{q}$ and
$q$ admit a section then $m\equiv 0, (n-1)^2\ \textrm{mod}\ n(n-1)$. Moreover,
using geometric constructions, we show that the homomorphism $\bar{q}$ and the
fibration $q$ admit a section for $m=kn(2n-1)(2n-2)$, where $ k\geq1$, and for
$m=2n(n-1)$.
  In addition, we show that for $m\geq3$,
$B_m(\mathbb{R}P^2\setminus\{x_1,\dots,x_n\})$ is not residually nilpotent and
for $m\geq 5$, it is not residually solvable.
