He teaches children at home in return for a monthly payment; if the child is absent is he entitled to payment?

If the agreement with the child’s guardian is to pay a specific monthly payment and it was possible for the child to attend but he did not attend, then you are entitled to take the payment in all cases, whether he is present or absent, so long as his absence was not due to some shortcomings on your part, because the payment is due just for setting aside time for the child. 

But if the payment agreed upon between you is per class, such that for each class there is a set payment, and the time or place was not set aside for this particular child, or the child was not able to attend because of some reason that was your fault, then you are not entitled to payment except for the classes that you actually taught. In that case you have to give back the payment for the classes during which the child was absent. 

The basic principle is that payment is dependent on work done. 

Payment is also according to agreements and conditions. It was narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “The Muslims are bound by their conditions.” Narrated by Abu Dawood, 3594; classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

Al-Bayhaqi (14826) narrated that ‘Abd al-Rahmaan ibn Ghanm narrated that ‘Umar (may Allah be pleased with him) said: Dues are based on conditions.

Classed as saheeh by al-Albaani in al-Irwa’, 6/303 

Shaykh Muhammad al-Hasan ould al-Duddo was asked about a worker who was hired to do some work and did not do it; is it permissible for him to be paid for it? 

He replied: 

If the person who was hired is available for work but the one who hired him turned him away, then he is entitled to payment. But if he was negligent and did not do the work that he was hired to do, then it is not permissible for him to take payment in return for that. This includes renting houses and apartments and also hiring people. If someone hires a worker for one month, then after he works with him for two or three days he no longer has any need of him, or he wants to move from that city, for example, then that worker is entitled to one month’s payment in full, because he made himself available, but the one who hired him no longer wanted him. But if the worker did not do the work properly and was not sincere, he is not entitled to the payment, because he is the one who did wrong. The same applies to one who rents an apartment for one month, then decides he does not need it and wants to leave it. The landlord is entitled to a month’s payment in full, but if the apartment was not as specified, then the landlord does not deserve the payment. End quote. 

Link to the fatwa (in Arabic):

www.islamway.com/?iw_s=Fatawa&iw_a=view

To sum up:

In the case mentioned in the question, you are entitled to one month’s payment in full, even if the student missed some lessons. 

And Allah knows best.