The German Empire was essentially a federation of several different states, most of which were, themselves, monarchies. That is, except for the three Free and Hanseatic cities in the north, Bremen, Hamburg, and Lübeck. They had been Republics and remained so in the Empire, so how did they coexist with and within the monarchies? Did the constution grant them any special privileges that exempt them from the influence of the Emperor? In what ways did they differ from the other states in interacting with the central government?
(Presuming you mean the the "Second Reich", 1871-1918, and not the Holy Roman Empire):
In my copy of the German Reichsverfassung (ie. Constitution, 9th print, 1904 edition, Guttenthag Berlin) the foreword explains that the prehistory is such that the Northern German Federation (Prussian dominated) already had Hamburg, Lübeck and Bremen join as independent city states in 1866; it was the rules of the Northern German Federation that basically became the blueprint of the Reich constitution of 1871. There were no "special privileges" insofar as their rights as states were concerned: like Bavaria and Saxony, they had a set number of delegates in both the Bundesrath (= upper House, 1 seat each; Prussia had 17, Bavaria 6, etc etc; there were 58 seats altogether) and the Reichstag (Parliament), for which there were regular free elections for a five year period (297 seats in 1904).
The Emperor did not as such have any more direct influence on a single German state than on any other; Hamburg was special only insofar as the special status of its Freihafen (Custom free international free harbor) needed specific constitutional regulation. Each German state had their respective constitution, election and parliament in some form, except that for the three Free Hanseatic Cities there was no constitutional monarch but a mayor (Bürgermeister) as head of state. These would send their number of deputies to the Bundesrath, accordingly.
The Emperor was summary head of state, needed to approve or could call for any dissolution of either Bundesrath and Reichstag, agree to declarations of war and all international treaties, and could pick the Reich Chancellor (who was automatically presiding the Bundesrath). (Note: the modern German spelling is Bundesrat, but to distinguish it from later forms of it I stick to the 19th century spelling here).
As such, neither of the 3 Hanseatic Republics were to my knowledge in any way different from the other German states.