How should one acknowledge a funding agency if it fulfils its obligations only partially? I applied for funding of my project and it was approved.

When it came to spending money, this funding agency put constraints on some sorts of purchases (e.g. by putting unrealistically small limits for computer buys). This was not stated in the call for proposals. 

The research along the proposed lines was done, with some degree of external support. How should one acknowledge the original funding agency provided that a significant degree of funds were not spent due to those suddenly introduced constraints and limits?
Since you did get support from your funding body, you will need to list it in the funding acknowledgements. This doesn't change with the fact that you have also gotten support from elsewhere. As profmartinez wrote, the proper term is to state that they provided partial support. Some researchers write "partly" exclusively in their funding acknowledgements, as their home institutions provide some level of support as well (offices, ...), which are not normally acknowledged.

The fact that your funding source provided less support than expected should not be written into the funding acknowledgements. Mentioning this somewhere would in the best case have no effect. In the worst case, it makes a bad impression both on you and the funding body. You have no ethical obligation to mention the quirks of their funding scheme, and under the "choose your battles wisely" premise, not mentioning the problem is the way to go.