Is it obligatory to perform ghusl after every intercourse in order to pray, especially since frequent bathing causes illness? Or is it permissible to pray in a state of janabah in this case?

It is not permissible to be lenient in abandoning prayer or performing it without purity. One must perform ghusl (ritual bathing) from janabah (state of ritual impurity) when intending to pray, unless ghusl would cause harm, in which case tayammum (dry ablution) is permissible by striking the earth and wiping the face and hands. Prayers that were performed incorrectly must be made up.