Conversion of magnitudes to Jansky and MAGPHYS?

I'm a bit puzzled and not an observer, so please bear with me if I'm being stupid here.

The code MAGPHYS specifies Jansky (Jy) as the input unit for flux through a filter (see Section 3.2.3 in the documentation). Meaning MAGPHYS wants me to use Jy as unit, when specify the total light of a whole galaxy in e.g. the SDSS-g band. But Jy is a unit of spectral density, i.e. W/m²/Hz. I would expect a flux as input unit when talking about the amont of light coming through a filter, i.e. in W/m² or in $L_\odot$ (solar luminosity), so integrated over all frequencies that go through the filter and weighted by the filter response function.

If I have the absolute AB magnitude of a galaxy in the SDSS-g band, how do I convert it to Jy so that MAGPHYS will be happy?

I'm aware of:

$m_{\text{AB}\nu} = -2.5 \log_{10} \left(\frac{f_{\nu}}{3631\text{Jy}}\right) \qquad \text{or} \qquad \left(\frac{f_{\nu}}{\text{Jy}}\right) = 10^{-0.4\ (m_{\text{AB}\nu} - 8.9)}$

But that is in one frequency, not a whole band.

Additionally, I have found the formula below in TOPCAT:

$\left(\frac{F}{\text{Jy}}\right) = 10^{\left(23-0.4 \ (m_\text{AB}+48.6)\right)}$

Where does the 23 in the exponent come from?

The input asks you for the flux through the filter, but a filter has a bandpass, so if you have a 10 nm bandpass filter and you measure $1 W/m^2$, then you measured $0.1 W/m^2/nm$ (sorry, I'm a physicist so I never got used to measuring wavelengths in Hz).
As for your second question, the factor of 23 is because in cgs units a Jansky is $10^{23} erg/s/Hz/cm^{-1}$ and that TOPCAT equation is in cgs (the equation is $m_\text{AB} = -\frac{5}{2}\,log_{10}\,f_v - 48.6$) and if you are normalizing it to Janskys, you need to divide by $10^{-23}$.