1. their

   • "With their relatives" refers to the family members.

2. most

   • "The cheapest and most interesting ways" fits because "most" is used to express the highest degree of interest.

3. channel

   • "My favourite channel is Cartoon Network" is the correct phrase when referring to a TV station or broadcast service.

4. watch

   • "Watch the programmes" makes sense in the context of viewing content on TV.

5. can

   • "My parents can also watch the programmes" is correct, expressing that it is possible for them to join.

6. On

• "On the website" is the appropriate preposition to use when referring to content available on a website.

So, the full text with the correct words is:

Nothing else to do. Most people watch TV in the evening because this allows them to sit in front of the TV set with (5) their relatives and to spend some free time together. As for me, watching TV is one of the cheapest and (6) most interesting ways for entertainment. My favourite (7) channel is Cartoon Network. It broadcasts all day so I can (8) watch the programmes whenever I have free time. It is a channel of entertainment for kids and families, so my parents (9) can also watch the programmes with me. I often search for my favourite programmes from Cartoon Network website. (10) On the website, there is a schedule for all programmes. I have relaxation time from watching this channel.