  We study the moduli space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ of
rank $r$ holomorphic bundles with trivial determinant and second Chern class
$c_2=k$, over the blowup $\tilde{\mathbb P}^2_{\!q}$ of the projective plane at
$q$ points, trivialized on a rational curve. We show that, for $k=1,2$, we have
a homotopy equivalence between $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ and
the degree $k$ component of the bar construction $\mathrm{B}\bigl(\mathfrak
M^r\mathbb P^2,(\mathfrak M^r\mathbb P^2)^{q},(\mathfrak M^r\tilde{\mathbb
P}_{\!1}^2)^{q}\bigr)$.
  The space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ is isomorphic to the
moduli space $\mathfrak M\mathcal I_k^r(X_q)$ of charge $k$ based $SU(r)$
instantons on a connected sum $X_q$ of $q$ copies of $\overline{\mathbb P^2}$
and we show that, for $k=1,2$, we have a homotopy equivalence between
$\mathfrak M\mathcal I_k^r(X_q\# X_s)$ and the degree $k$ component of
$\mathrm{B}\bigl(\mathfrak M\mathcal I^r(X_q),\mathfrak M\mathcal
I^r(S^4),\mathfrak M\mathcal I^r(X_s)\bigr)$. Analogous results hold in the
limit when $k\to\infty$.
  As an application we obtain upper bounds for the cokernel of the Atiyah-Jones
map in homology, in the rank-stable limit.
