True, the governors can remove proposed budgets.
There are only a number of states, commonwealths and territories that allows the governor this power of reduction where he can remove certain portions of a proposed budget. The term "reduction" mostly preferred as "line-item" is a part of veto power which can be used to remove any appropriation that is objected to by them. This practice allows the governor and the budget staff plays a very crucial role in making priorities and taking decisions accordingly by using the state resources.
The given statement “many governors have the power of reduction, allowing them to remove certain portions of a proposed budget” is true.
The U.S Constitution empowers the governor to interfere in the budget framed by the state legislators and remove the unnecessary provisions if he finds any to be included. This is primarily done to reduce the burden on the treasury of the state and cease any unnecessary expenditure. The power to do this is denoted as 'reduction veto' of the governor.
The governor manages the executive authority of the government in each state or territory. Then, in addition, based on the jurisdiction, has significant control over the government budget, the power to appoint many officials (includes judges), and an important role in legislation.