The situation you are in is called an "Escrow Shortage". This means that due to an increase in either your property taxes and/or your home owner's insurance, your mortgage payment is too low to pay for principal, interest, taxes, and insurance. The bank is collecting more money from you to cover the taxes and insurance in your payment as requested when you signed your loan.
There are three options:
1.) Allow the lender to recalculate your mortgage payment. This means the lender will add more to your payment to (a) cover the new, higher cost of property taxes and/or insurance PLUS (b) collect enough extra money to make up for the shortage in payments you've so far this year. It is like a double whammy.
2.) Ask the lender if you can add to your escrow account to cover the shortage this year. As you've asked, you would be able to put some cash into your escrow account to pay the difference. However, your mortgage payment will still be higher - enough to cover the new higher insurance and/or property taxes
3.) If the increase is due to insurance only, shop for a new policy. You will need to know the minimum coverage required by the lender, the maximum deductions the lender will allow, and which flood plain your home is located in. Your lender's customer service dept. can tell you minimum coverage and maximum deductible. Your appraisal will have the rest of the info an insurance agent needs.
If the increase is due to an increase in property taxes, you have few choices but to accept the higher payment. It MIGHT be possible to appeal to your local tax assesor for a reduction IF you can document that your home has fallen in value.
Try paying the difference in escrow first. That will give you the best shot at minimizing the amount of your payment increase.