Mr Mallard's colleague Ruth Dyson also voiced disappointment at Ms Turei's decision to lodge the amendment, saying there had been unanimous agreement on it in the committee after extensive consideration.
Ms Turei said the Greens had made it clear they did not agree with the decision to allow the Speaker to keep control of travel, but supported the bill as a whole as it made progress toward ensuring MPs' entitlements were decided more independently.
Ms Turei said she had argued against it during the select committee's deliberations, and had put in the amendment so that Parliament as a whole could make the final decision, rather than the select committee.
MPs now have unlimited domestic travel at the taxpayers' expense. The personal benefit of this is calculated by the Remuneration Authority which takes it into account when setting the base salary.
The bill was sparked by a Law Commission review and followed years of expenses scandals.
Attorney-General Chris Finlayson said there were submissions concerned about handing travel over to the Remuneration Authority - a reference to Clerk of the House Mary Harris' submission, who said Parliament was supposed to be in charge of its own operations and travel was a critical part of being an MP.
Mr Finlayson said keeping that decision with the Speaker was to address that concern. Travel by MPs' families would be set by the authority because it did not jeopardise MPs' ability to carry out their role.
Proposed salary increase has MPs squirming
Members of Parliament are again squirming over their salaries after news the Remuneration Authority is considering a pay increase of about 2 per cent - and possibly more - for them this year partly in a bid to help bring ministers' salaries closer to those paid to public service heads.
PM John Key revealed it had come up with its proposed increase this week and said he had written back saying MPs should only have a minimal or zero increase.
The authority has indicated it is looking at about 2 per cent - which would increase a backbench MP's salary by $2,826 to $144,636 and the Prime Minister's by more than $8,000 to $419,740. A Cabinet minister would gain more than $5,000 to be making $263,000.
Yesterday, other MPs distanced themselves from the decision which is expected to be announced midway through next month.
Labour leader David Cunliffe said it should be left to the Remuneration Authority to decide without interference from MPs.
In National, Foreign Minister Murray McCully said he agreed with Mr Key's stance, and Chester Borrows said that there were others going without pay increases "and so should we".
Labour MP Shane Jones said the decision was up to the authority, "but don't underestimate how long and hard the hours are that your MPs are toiling away".
Labour's Clare Curran said that with the low pay increases many workers faced this year, "I think it would be counter-productive for MPs to be given a pay increase."