= Solution
Adding $40000$ monovalent <counterions> makes the combined genome-plus-ion <electric charge> nominally zero. It does \b[not] make the local <electric field> or <free energy> vanish: opposite charges have spatial distributions, and confining mobile <ions> costs <entropy>.
A definite screened continuation of <4/b> replaces the bare <Coulomb potential> by $e^{-\kappa r}/(4\pi\epsilon r)$, with $\kappa=1/\lambda_D$ and $\lambda_D=1\,\mathrm{nm}$. In the <Debye–Hückel approximation>, a uniformly distributed fixed <charge density> $\rho=Q/V_c$ satisfies $(\nabla^2-\kappa^2)\varphi=-\rho/\epsilon$ inside, with zero fixed charge outside and matching potential and radial field at $R$. Its interior solution is
$$
\varphi(r)=\frac{\rho}{\epsilon\kappa^2}\left[1-(1+x)e^{-x}\frac{\sinh(\kappa r)}{\kappa r}\right],\qquad x=\kappa R.
$$
The screened charging <free energy> $U_\kappa=\tfrac12\int\rho\varphi\,dV$ is therefore
$$
\boxed{U_\kappa=\frac{Q^2\lambda_D^2}{2\epsilon V_c}
\left\{1-\frac{3(x+1)}{2x^3}\left[(x-1)+(x+1)e^{-2x}\right]\right\}.}
$$
For $R\gg\lambda_D$, the bracket approaches one: only charges within roughly a screening length contribute appreciably, yielding the local estimate $U_\kappa\simeq Q^2\lambda_D^2/(2\epsilon V_c)$. With the same $\epsilon_r=80$ as before, $x=20$ gives bracket $0.92519$ and
$$
\boxed{U_\kappa\simeq8.00\times10^5\,\mathrm{pN\,nm},}
$$
about $173$ times smaller than the unscreened packed-sphere self-energy. This is a specified continuum estimate of packed-state charging <free energy>; a full packaging <free energy> also subtracts the screened unpacked reference and includes ion redistribution. It should not be presented as a unique exact packaging <work> fixed by the problem's few inputs.
The supplied <counterion> count also permits an independent estimate of the entropic contribution. For $N$ ideal confined <ions>, $F_{\rm ion}=-Nk_BT\ln V+\text{constant}$, so compression from an accessible reference <volume> $V_0$ to $V_f$ costs
$$
\boxed{\Delta F_{\rm ion}=Nk_BT\ln\frac{V_0}{V_f},\qquad Nk_BT=1.66\times10^5\,\mathrm{pN\,nm}\quad(T=300\,\mathrm K).}
$$
For example, approximate the original screening atmosphere as an annulus of thickness $\lambda_D$ around a rod. With $a=0.5\,\mathrm{nm}$, $V_0=\pi[(a+\lambda_D)^2-a^2]L=4.27\times10^4\,\mathrm{nm^3}$. Taking $V_f=V_c-V_D=1.35\times10^4\,\mathrm{nm^3}$ gives $\Delta F_{\rm ion}\simeq1.91\times10^5\,\mathrm{pN\,nm}$. This useful order-of-magnitude ion-entropy model is distinct from the screened charging model; the two estimates must not simply be added as independent terms without a consistent ionic <free energy> functional. The rod cutoff, fixed <DNA> volume and screening-atmosphere geometry are rough geometric assumptions, not a microscopic reconstruction of the source sketch.
There is a further consistency limit. The nominal ion <number density> is $n=N/V_c=1.19\,\mathrm{nm^{-3}}$. For a uniform ideal monovalent screening gas, $\lambda_D^2=\epsilon k_BT/(ne^2)$ would give $\lambda_D\simeq0.309\,\mathrm{nm}$ at $300\,\mathrm K$, rather than $1\,\mathrm{nm}$; added salt would shorten it further. Thus the stated ion inventory and screening length cannot both be interpreted as a self-consistent dilute uniform screening gas with this dielectric. Bound <counterions>, nonuniform distributions and an effective phenomenological screening length can change that interpretation. Linear screening is itself only an estimate near highly charged <DNA>.
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