The following response was adapted with permission from You Can Teach's TESOL course: You Can Teach TESOL and Recruitment in China : Home
You are in a tough spot. I taught in China for many years, and taught classes of up to 60 students while in Harbin (you may laugh at this small group!). I was in much the same position as you; only seeing them once a week and required to 'improve their speaking and listening'. I agree with you it is nigh-on-impossible to make any real progress, and your time would be much better spent on reading instruction (which requires speaking and listening skills to be used, and ultimately improves them too). However, there are some things you can be doing to at least make sure they are getting SOMETHING from your classes.
My advice is to divide and delegate. There is no way you can give all of these students a chance to practice (discounting drills) unless you divide the class into smaller groups. By dividing the class into groups and allowing students to be the leader of each, you can have each group completing a game/activity simultaneously, while you move around keeping students on task and answering questions. The size of the groups will be determined by the nature of the activity, and the number of above average students you have (see below).
The next thing you need to do is assign each group a leader, and delegate some of your authority to them. The main determinant of who you make leader should of course be their level of English, but behaviour and standing amongst peers should also be considered. These guys will essentially be peer-teachers.
Before we look at some activities, it is important to remember that things will get noisy! However, as long as they are speaking English as they make noise, and quiet down when you ask, this is nothing to worry about.
Chinese whispers - great game you probably know well. Students don't need to move around, as they can simply whisper the word to the person nearest. I would suggest eight groups for this one, or just one group per column. You whisper the sentence (a variation of the target language obviously) to the students in the front row, then on your signal they whisper it to the back. Students at the back walk to the front of the class and tell you what they hear. Your group leaders should not participate, but rather help you manage the other students.
Word Division - A great game for listening/reading/speaking practice. Students are each given a small piece of paper with a word written on it (again, use your leaders to help you distribute). These words make part of the sentence you are teaching. For example, if you are teaching 'What time is it?" you would have 20x 'what', 20x 'time' etc. The words are shuffled before distribution. When each student has a word-card, you will call out a sentence with one word missing, e.g. "What blahblah is it?", and all students with 'time' need to stand up and shout 'time'! then have the whole class repeat the question, or have them say each word in turn (i.e. 20 kids stand up and say "What", next 20 "Time" and so on). Then have the students swap their cards and repeat.
Running a bit short on time and can't seem to think of any more activities at this point! Basically you can use any game/activity you come across. Just find some trustworthy students to help you manage them, and don't be afraid of a little noise!
Hope this helps!