In short, yes. You can set off any VAT which you are charged against the VAT which you, as a registered supplier, have to pay to HM Revenue & Customs each quarter.
If you suffer more VAT than you charge (although that is unlikely in a ‘standard rated’ business) then you can claim a refund from HMRC.
It’s best to keep a separate set of accounts for all your VAT figures, and the extra cash you are now paying to your quantity surveyor will simply reduce the cash you have to send to HMRC at the quarter end.
None of it has any direct effect on your trading profitability (although of course it would be easier to get jobs if you were not having to charge VAT on your own prices), and it can be a good discipline not to think of any VAT (in or out) as your own money; it’s all HMRC’s.
The only complication is if you are ‘partially exempt’, which is generally more of an issue for dentists, for example, and other healthcare providers. They cannot reclaim VAT suffered as part of making an exempt supply, and have to apportion all their VAT claims according to the ratio of exempt to VATable sales.
That shouldn’t be an issue for you, but if you have any worries about how best to handle your VAT then talk to a qualified professional – local recommendations from other construction firms would be best, as they’ll know about CIS as well as VAT.