Can I apply for a patent for this method?

This method can be patented. But it must meet the three characteristics of patent, namely "novelty, creativity and practicality".

For example, the manufacturing method of a certain product and so on.

Method patents are usually used in the manufacturing process of products. For example, a well-known product can no longer meet the requirements of applying for product patents, but you have invented a new production method, which has the advantages of significantly increasing output and reducing costs, and so on. This method can be used to apply for a method patent.

One of the characteristics of method patents is that it is difficult to obtain evidence when encountering patent infringement disputes. Because you can buy products with patent infringement through normal channels, it is difficult for you to know the production methods used by the infringer through normal channels.