How does GDP get manipulated by government budget decisions?
There are at least a couple of ways that you could understand this question. First, it might be asking how the government directly manipulates GDP. In other words, this might be asking how government budget decisions lead directly to a change in GDP. If so, the answer is that the government cannot really manipulate GDP in any major way. It is remotely possible that the government could increase the GDP in a given year by doing a lot of government spending in that year. The government could buy things in a given year that it would normally put off until the next year. This would artificially inflate GDP numbers for the current year, but of course it would reduce GDP in the next year. This is not something that is commonly done.
Second, the question might be asking how government budget decisions can have an impact (albeit indirectly) on GDP. The answer to this is that the government often tries very hard to change GDP through budgetary decisions. This is called fiscal policy. When the country is in a recession, for example, the government typically increases spending. This puts more money in the hand of consumers who go out and spend it. This helps increase GDP, or at least keeps it from falling as fast as it might otherwise do. If we include taxes as part of government budget decisions, we can say that the government sometimes reduces taxes to try to increase GDP. When it reduces taxes, people have more money and can spend more on goods and services. This increases GDP (or keeps it from falling).
It is not completely clear to me which of these types of manipulation you are asking about. The second type is much more common than the first.