What "Gate" name in a historically important North-English town does not refer to a city-gate, but to a little street where public flogging and beating was practised?

well I'm not sure if this is the one you're looking for, but you have to be bemused by the name of this street in (the historically important northern English town of) York. The street name is Whipmawhopmagate.
reference:
https://www.tripadvisor.co.uk/Attraction_Review-g186346-d19392644-Reviews-Whip_Ma_Whop_Ma_Gate-York_North_Yorkshire_England.html

It originally seems to have been called the "whit-nour-what-nour gate" which is also a rather odd streetname.
https://en.wikipedia.org/wiki/Whip-Ma-Whop-Ma_Gate

It should be noted that -gate in the North of England at least doesn't usually refer to a city gate. Roads like Longmeanygate and Taylor's Meanygate have never been urban enough for any more than five-barred gates. The -gate means road or street, so when by itself as in Northgate it will mean North Street; Northgate Street is likely to lead to the North Gate. Gasse is the German equivalent linguistically.

Actually, in Norway a street is called a "gate", but it's not pronounced like in English.

According to this source, the suffix -gate in England often refers to the name of a Roman road.

https://en.wikipedia.org/wiki/Gate_(disambiguation)