Nope
Either you give the coat of arms (a city seal if you removed the lettering around it and enlarged what was in the middle let's say and could pretend to be the city coat of arms) as a symbol of the city or you give the colors alone with simple symbolism do not combine the two.
A good example of these flags that I like are:
The unofficial flag of Radom, formally Radom does not have a city flag and only uses either the coat of arms or the county flag.
The county flag used as a substitute for the city flag.
No combination, the flag of Krakow. Two stripes, simple and pleasing to the eye.
Here another simple example from Warsaw, no fancy decorations. Just the colors of the coat of arms transferred to the flag, the same as with the Kraków flag. BTW, the flag of Poland is also the colors of the coat of arms transferred to the flag.
One last example, the flag of Gdansk. Coat of arms symbols transferred to the flag which has the color of the coat of arms shield.
Why do you Americans have to make life so complicated for yourselves and create this thing that can barely be called a city flag.
After all, it is not difficult, just two random even colors that are associated with the city, some symbol of the region and boom. You have a simple and pleasing to the eye flag.
As a last resort, you can always steal the colors from a well-known city sports team.
Or dig out some old painting depicting uniforms of soldiers coming from that city and give the colors of the flag on their colors. (Yes BTW the flag of Plock was made, an old legal act from 1792 was pulled out that the colors of the uniforms of the province of Plock were to be such and such, and such a flag was made).
But smaller, what other bad flags do you have that you might want to murder the authors of this abomination?