Learn English – What’s the meaning of “on somebody’s account”

Henry's answer is exactly right. Here is confirmation:

A clearer, more complete use of "cutting" comes later in the same paragraph.

He wasn't willing to cut a subsidy for heating bills.

This illustrates that "cutting," as used in your example, had an object (parts of the proposed budget), and was not just used as an expression meaning something else.

The sentence in your question might be paraphrased in this way, for the sake of this explanation: "As the government programs fell and the tens of billions in savings piled up during the budget debate in the House last week, freshman Republican Frank Guinta joined fellow congressmen in reducing spending and removing parts of the proposed budget.

The pattern

X modulo Y

is an informal but common parlance in technical, especially mathematically, oriented talk. It is used to mean informally 'X, ignoring Y'. For example,

"The rocket design was flawless, modulo the toxic waste produced by its fuel."

The meaning is inspired by, but not perfectly corresponding to, the arithmetic modulo function (for example, clock-time addition) which when suitably abstracted involves 'collapsing' all items of a set into the special items of the set, so that the full set does not need to be dealt with (this is where the associated meaning of 'ignoring' comes from).

In your interpretation "note what parts of it still need to be modified", the 'modified' part is irrelevant. 'Modulo' is pragmatically "I'm telling you about the most important part (the X), but remarking on the existence of some part that might be important for other reasons but under the current context we want to ignore (the Y)".