This error is becoming quite common, due to Windows applications such as git and rsync that are based on Cygwin. Have you followed the instructions? In general, your third party applications should work with the current Cygwin. Some very old applications may need updating.
If you can’t update the third party applications that install cygwin1.dll you can try configuring cygwin to omit the Windows system PATH. From Cygiwn’s /etc/profile:
[...] setting CYGWIN_NOWINPATH non-empty in the system variables
prevents use of the existing PATH and a clean PATH just for
Cygwin is set up [...] add any extra path components
you need in your personal startup files