The request of your product manager really means that she/he wants to review your plan.
The fact that the team missed deadlines already means, that something is wrong on the estimation, planning or execution side of the project.
It is normal to plan for the activities of each sprint in details in advance.
Every task in your sprints should have a complexity and cost factor associated with it.
If I were the manager I would be interested to learn if adding another resource will really solve the issue and delivery will happen as planned. The best way to see this is to look at the plan, identify the issues and bottlenecks.
There could be several things wrong with a project plan. All tasks may be underestimated due to lack of information and gray areas during planning, another reason may be some hidden complexity discovered during the project, or a steep learning curve, lack of skills, uncooperative third party teams, unexpected dependencies and I could go on.
Most managers drive based on facts, and this is exactly what they get with the plan review. This is where you can explain where the issues are and wether adding more people is the right step.
I don't see another way to convince your manager after she/he explicitly asked for these facts.