I must admit that I was very touched by your post, because I have very similar experiences.
I really feel sorry for the four year old boy and I was stunned that he showed his bad mood even in front of you. Children of his age do not normally react like this: they usually become a little bit shy when they meet you for the first time, but observe you and are curious to find out who you are. It occurred to me that he already suffers from the pressure his parents put on him.
I think that you are definitively right to make friends with the little boy, play with him and accompany him. That is a very good approach and I am sure you will have very good results soon.
I am an English teacher at Elementary School and often asked to teach very difficult children individually. I f.e. use this page Home English Teacher
(There is also an Amazon paperback if you want) which includes a lot of playful encouraging activities and there are also wonderful videos. They show how to built up a good relation with the child and how the child becomes more and more confident: children become keen to practise the language because they have fun and have a good and trusting relationship with the tutor. I have made very good experiences with this material and children who are said to be difficult always get very involved when we do the activities.
I also think you should study some literature so that you can defend your approach against the parents and the agency who hired you. You should be able to argue why playing games is important in teaching children so that they will not critisize or blame you.
I can recommend studying Blooms taxonomies - here is one example for numerous pages: Bloom's Taxonomy of Learning Domains
. Bloom is a famous scientist whose work has inspired the conception of many school curricula in the USA and in Europe and his taxonomies already have influenced generations of teachers in their lesson planning. I have found that didactical games and language games strongly refer to Blooms taxonomies of knowing, understanding, applying, evaluating and creating and therefore can create a good basis for the child´s learning.
If you want to find out more about using games in teaching - because you said you are also going to tutor the 11 year old sister - you can use ESL Games, Stories, Songs and Plays for Kids, Teens and Adults!
. Here you can find a lot about the benefical impact of games on learning together with detailed methodological advice. There is also a huge bank of all sorts of games which refer different taxonomies of thinking as well as on all language skills ( listening, writing, speaking). You will also find inspiration for teaching children with learning difficulties.