What do you advise me to do to be able to conceive, and is an expiation for breaking an oath (kaffarat al-yamin) incumbent upon me, given that I swore not to have a third child until my second child enters the first grade, which has not happened yet?

Expiation (kaffarah) becomes obligatory if you break your oath. For instance, if the questioner gives birth before her young son enters the first grade, then if her intention was giving birth (childbirth), then the oath is broken by birth. If her intention was conceiving (getting pregnant), then the breaking of the oath occurs upon conception. As for her mere desire to give birth and her determination to do so, this is not considered breaking the oath, and no expiation is required for it. Expiating an oath before breaking it is permissible according to the majority of scholars.