What must a pregnant woman do to preserve the health of her fetus, and are there specific adhkar (supplications/remembrances of God) to be recited throughout the pregnancy?

There are no specific supplications for pregnant women or fetuses in the sound Sunnah. What has been narrated in this regard is either fabricated or an ijtihad (independent reasoning) from some scholars. Among the fabricated narrations is the mention that the Prophet, peace and blessings be upon him, ordered the recitation of Ayat al-Kursi and the two Mu'awwidhatain (Surahs al-Falaq and al-Nas) when Fatimah was born. Examples of ijtihad include what Sheikh Ibn Uthaymeen mentioned regarding reciting verses that indicate ease, such as: (Allah intends ease for you and does not intend hardship for you), and (No female conceives nor delivers except with His knowledge), and (When the earth is shaken with its [final] earthquake, And the earth discharges its burdens); for the entire Quran is a healing. And the supplication of a mother for her child is answered, so it is hoped that her supplication for her fetus will be accepted for safety, righteousness, guidance, success, and kindness to his parents.