ネットワークエラーと関係するかはわからんが、Wine5.7以降はCAP_NET_RAWの有効化が必要とか。

https://wiki.winehq.org/Debian

Beginning with Wine 5.7, the WineHQ Debian packages have an optional debconf setting to enable CAP_NET_RAW to allow applications that need to send and receive raw IP packets to do so. This is disabled by default because it carries a potential security risk, and the vast majority of applications do not need that capability. Users of applications that do need it can enable CAP_NET_RAW after installing Wine by running
dpkg-reconfigure wine-<branch>-amd64 wine-<branch> wine-<branch>-i386
and answering yes to the three questions. (Substitute devel, staging, or stable for <branch> in the above command.)