The question is about identifying which type of engineering contract involves payment based on the actual quantities of work done at pre-agreed rates. Let's discuss the given options:

1. Lump Sum Contract: In a lump sum contract, the contractor agrees to complete the whole specified work for a fixed sum of money. Payment is not based on quantities of work done but rather on the completion of the entire project or specific milestones. This option is not the correct answer.
2. Turnkey Contract: In a turnkey contract, the contractor is responsible for both designing and constructing the project and handing it over to the employer ready for operation, often for a fixed price. Like the lump sum contract, payment is not primarily based on actual quantities of work done, but rather on the completion of the project. Therefore, this option is incorrect.
3. Cost-Plus Contract: In this type of contract, the contractor is paid for all construction-related expenses plus an additional payment to allow for profit. This payment structure is based on the actual cost of the materials and labor and not on a schedule of rates. Hence, this is not the correct choice.
4. Item Rate Contract: In this contract, each item of work is assigned a rate, and payment is made on the basis of actual quantities of work accomplished. This means the contractor is paid for the specific quantities of work completed, evaluated at previously agreed ratess. Therefore, the most suitable option for the given question is the Item Rate Contract.

Therefore, the correct answer is Item rate contract, as it involves payment based on the actual quantities of work done at agreed rates.