make[2]: Leaving directory `/home/freeciv-2.4.0-beta1/R2_4_0_beta1/dependencies'
Making all in utility
make[2]: Entering directory `/home/freeciv-2.4.0-beta1/R2_4_0_beta1/utility'
cd . && ./generate_specenum.py
/usr/bin/env: python: No such file or directory
make[2]: *** [specenum_generate] Error 127
Obvious question (I haven't looked into this in any depth): do you have a python executable as part of your MinGW installation?
(If not, perhaps configure should have spotted this. If so, perhaps we have a problem finding it, or making the specenum script use the found one. In any case, I don't see anything mentioning Python as a build dependency in the INSTALL instructions.)
Python should not be dependency when building from release tarball. Tarball should contain all those files python is needed to (re)generate.
Have you copied files around or maybe some other way messed with file timestamps? That could cause it to think that some files are outdated, and should be regenerated with python. What source package exactly are you using (.tar.bz2 or .zip?) OTOH if you build from svn checkout/export, you definitely need python.
If you are using svn anyway, you may want to consider using head of stable 2.4 branch (S2_4) instead of beta1 tag. There's quite a number of bugs fixed since beta1 http://freeciv.wikia.com/wiki/NEWS-2.4.0-beta2