Republican house has lots of power, If they are unified.

 THE POWER OF NO IS THIER POWER

THEY CAN FASHION THIER OWN BUDGET AND PROCEDURE

While the Republican house has a very slim majority, it can still have lots of power when it come to spending. The item that should be of the utmost priority. They can refuse to vote for any new spending. They can break up omnibus spending bills and vote and amend them as they see fit. They do indeed have the power of the purse, if they have the unity and the backbone to use it

They should be working on budget items every day, passing them after they have amended them, and send them to the Senate. They can vote on individual items and stop the habit of large bills laden with pork.

When the ominous deadlines keep arising, they can continue to pass continuing resolutions, that in effect will freeze the budget at existing levels.

They should be very active, passing funding measures every day and leave it up to the Senate and the President to either pass them or say No.

They do not have to threaten to shut down the government, they should always have a bill to fund the government at least at the present level. 

This is the only power they have. They cannot pass any new legislation. They need to only focus on what they can and that is to cut pork barrel spending or at least freeze spending at current levels. Then they are doing their job and let the results speak for them.