can we put the cost of the lien in the lien or the back finance charges

That's a great question. New York, like many states, limits the amount of a lien claim to the amount that's owed but unpaid for work performed. By including amounts that exceed what's owed for work performed and unpaid, a lien claimant could actually run the chance of their lien being considered exaggerated - and that could lead to the loss of the claim and potentially further liability. Of course, if the dispute ultimately does result in legal action, it's always possible for a lien claimant to recover things like attorney fees, filing costs and fees, and interest payments. Further, when negotiating the release of the lien claim, it's always possible for the claimant to try and compel payment of things like filing costs and interest fees. We've discussed the issue on the zlien blog: New York Rules On What Is A Fictitious Lien. For more information on New York's mechanics lien rules, this resource should be valuable: New York Lien & Notice Overview.