The correct way to perform ruqyah for a small child in order to protect him is what the Prophet (blessings and peace of Allah be upon him) used to do for his (grand)sons al-Hasan and al-Husayn (may Allah be pleased with them).
Al-Bukhaari (3371) narrated that Ibn ‘Abbaas (may Allah be pleased with him) said: The Prophet (blessings and peace of Allah be upon him) used to seek refuge with Allah for al-Hasan and al-Husayn, and he said: “Your father (Ibraaheem) used to seek refuge with (these words) for Ismaa‘eel and Ishaaq: A‘oodhu bi kalimaat Illaahi’t-taammah min kulli shaytaanin wa haammah wa min kulli ‘aynin laammah (I seek refuge in the perfect words of Allaah from every devil and every vermin, and from every bad eye)”.
Ibn Hajar said in Fath al-Baari (6/410):
The word “haammah (vermin)” refers to poisonous things.
The words “from every bad eye” – al-Khattaabi said: What is meant by this is every disease and ailment that may affect a person of insanity and confusion. End quote.
It is also mustahabb when performing ruqyah for children to recite al-Mu‘awwidhatayn (the last two soorahs of the Qur’an) over them and to wipe their bodies whilst reciting, or to recite them into one’s hands and blow in them, spitting lightly, then wipe them over whatever of their body one can reach, or to recite them into water and wipe them or wash them with it. The Prophet (blessings and peace of Allah be upon him) used to seek refuge for himself and others by means of these two soorahs.
It was narrated that Abu Sa‘eed al-Khudri (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) used to seek refuge with Allah from the jinn and people’s eyes (i.e., the evil eye) until the Mu‘aawidhatayn were revealed, then when they were revealed he began to recite them for protection and stopped reciting anything else.
Narrated by at-Tirmidhi (2058); classed as saheeh by al-Albaani in Saheeh at-Tirmidhi.
Blowing with a little saliva after reciting the Mu‘aawidhatayn is taken from the practice of the Prophet (blessings and peace of Allah be upon him) just before going to sleep. He used to recite them into his hands and blow into them, then wipe his body with his hands. When he fell sick, ‘Aa’ishah used to do that for him, which indicates that in the case of a small child, his mother can recite the Mu‘aawidhatayn and blow (into her hands) and wipe his body with them.
It was narrated that ‘Aa’ishah (may Allah be pleased with her) said: When the Messenger of Allah (blessings and peace of Allah be upon him) went to his bed, he would blow into his hands after reciting Qul Huwa Allaahu Ahad (Soorat al-Ikhlaas) and the Mu‘aawidhatayn, then he would wipe his face and whatever he could reach of his body with his hands. ‘Aa’ishah said: And when he fell sick, he would tell me to do that for him.
Narrated by al-Bukhaari (5748).
With regard to the adkhaar for the morning and evening, there is no report – as far as we know – to suggest that they may be recited over others with the intention of performing ruqyah, so we do not advise you to carry on doing that; rather you should limit it to what has been proven from the Prophet (blessings and peace of Allah be upon him), for that is sufficient. The best of guidance is the guidance of Muhammad (blessings and peace of Allah be upon him).
And Allah knows best.