The relationship between state governors and the United States Congress can be a little murky at times. In the sense of having a direct role in the federal legislative process, state governors may be able to assert some political influence over their own state's senators and representatives, but they have no official role in the voting process. State governors have much more power and influence at the state government level.
The US Congress has an established pecking order, beginning with freshman or first year House representatives and ending with long-serving senators and the appointed president pro tem, usually a respected senator from the majority political party. State governors are not included in that hierarchy of power, although they are not without their uses for federal legislators.
Quite often state governors find themselves on the thankless receiving end of new federal laws, Supreme Court rulings and unfunded mandates imposed by Congress. Because these edicts are often tied to much-needed federal funding, state governments are generally obligated to comply with any federal laws, and in many cases are strongly urged to create similar state laws, such as the mandatory use of motorcycle helmets or highway speed limitations. Even if certain state governors believe the federal laws are either too restrictive or too permissive, they are rarely in a position to override federal law with new state law.
This is not to say that state governors are completely powerless when dealing with the United States Congress. Individual state governments can band together over an issue of mutual interest and use that collective power to send a unified message to Congress. State governments have historically chafed at the idea of being controlled by a strong central federal government, which in turn has led to several skirmishes over states' rights. The Civil War, for example, was partially triggered by a disagreement between state governors and Congress over the right to decide whether future states could decide to allow slavery or not.
Another infamous clash between a state governor and the federal government occurred in 1962, when Alabama governor George Wallace physically stood in the doorway of the University of Alabama to prevent several black students from registering at a traditionally white college. While the federal government had ordered the desegregation of government-funded institutions, Governor Wallace believed individual states should have the right to form their own segregation or desegregation laws. In the end, Wallace did step aside and allow the students to enter the campus, but the incident demonstrated how fractious the relationship between state governors and Congress can be.