If I understand you, you have daily data and you are making weekly forecasts and monthly forecasts and finding the monthly forecasts more accurate, so you believe that if you forecast by month and then try to disaggregate this into weekly values you'll have better results than if you forecast by week in the first place?
The monthly forecasts are better because they ignore weekly effects, smoothing/averaging things out. They have a lower variance because you're aggregating together more data and throwing away distinctions that are in the data. If you then try to pull weekly results back out, you've essentially taken a round-about way of smoothing your data and then doing weekly predictions.
As an example, if you're looking at business data and you don't appropriately model a holiday in a week, that will totally mess up that week's prediction. If you are modeling a month, the holiday's impact will be 1/4 or less (assuming four weeks in a month) than it is in a week.
Similarly, if the holiday doesn't always fall in the same week of the year, it may still fall in the same month, so it would not affect your monthly forecasts but it would add a lot of noise to weekly forecasts as it hops from week to week.
Last, you would also be using data from the future for your weekly forecast. Your weekly forecast in week 1 of a month will be based on data from all four weeks in the month, which you won't have in week 1 when you try to predict in the future.
Weekly forecasting is hard for a variety of reasons, which is why official government statistics are usually based on months or quarters. You may improve the results of a poorly-specified weekly model by disaggregating from a monthly model but that's only because you're smoothing over mistaken assumptions in the weekly model and you're pulling weekly data from future weeks to help you.
(I'm not even getting into the fact that there aren't a whole number of weeks in a year, so they'll never line up from year to year and seasonal effects will be wonky on a weekly scale. You really don't want to do weekly prediction if you can avoid it.)