That's kind of a loaded question, they work the the Governor detail, highways and transportation system ferries Etc., they will and do respond to city calls at time for back up, if a city does not have a police force, then the County Sheriff would most likely be the first responder.
Here in Thurston County WSP will respond to certain county roads in place of the sheriff if they are out of position/tied up on other calls. I can't say I've ever heard them respond to anything more than traffic issues (usually accidents) besides on really big events.
The accidents are broadcast to the units either first hand or second hand from the PSAP and it's the troopers discretion if they want to respond or not. Usually they do. I haven't heard 'certain roads' being an issue in the counties. They work Capitol Way a lot lately up to the city limits of Tumwater and Harrison and Division in Olympia looking for DUI's and the like. I saw one today pulling a motorist over on the Capitol Way viaduct over I-5. If they choose to patrol some city streets, it depends on what their emphasis is for the day.
Law enforcement here in the US is much different than in Canada. The RCMP have nationwide authority, which police in the US do no not have.
Back in the early 20th century when states were forming state law enforcement agencies, some were called "State Patrol" or "Highway Patrol" instead of "State Police", to somewhat restrict their authority.
Here in Ohio the state patrol pretty much stick to state highways and interstates. However in Kentucky, where they are "State Police", they may be the only police in areas with little or no law enforcement and will respond to any/all calls for police service.
The Washington State Patrol is a general authority police agency with full jurisdiction anywhere in the State except for federal reservations (military bases, tribal lands, etc) and have the ability to enforce any and all State laws anytime they want. In general the uniformed Troopers stick to the State highways but they can and do make stops anywhere, and respond to back up city and county officers on calls. When I was an officer in Sultan the WSP was frequently my only backup and they were great.
In the late 80's Washington State passed a law called the Mutual Aid Peace Officer Powers Act. That act gave authority for all police officers in Washington to enforce laws anywhere in Washington. The idea was to remove any issues for neighboring departments when they back each other up. It also means that a Spokane police officer can stop you in Tacoma.