Unfortunately poor company cultures exist. This is why Organisations That Matter was created - to try, one company at a time, to change this reality.
So, what can a Team Leader do to improve the motivation of their team members when their team exists within a poor company culture?
The first thing is to control what you can control as the Team Leader. My experience and research indicates that pay is a massive de-motivator if it is not 'fair' in the overall context of pay within the company and the industry that you are in. If a person is not being paid fairly in this context, then pretty much everything else that the company does becomes less relevant over time. The pay issue becomes the core de-motivational issue.
If pay is 'unfair' then you must do what you can to fix that situation.
Most companies have systems and processes for accurately paying people. As such, if you believe that a person is being paid unfairly, go in to 'bat' for them. Let them know what you are doing but also let them know that you have to follow the system's rules. Providing you have a record of being genuine, most people will be very pleased that you have taken the time and effort to go in to 'bat' for them. This act of support will often increase a person's motivation. However it won't last forever if the real problem isn't addressed. (Please note I accept that the research indicates that most people feel they are underpaid. However, when 'pay' is placed in the context of company and industry, it is my experience that most people are able to identify if they are paid within an acceptable 'range' of pay.)
If pay is 'fair' then it is the cultural issues that come into play. It is possible, within limits, to create a positive sub-culture that may exist only in your team.
From my research from conducting many leadership development activities on this exact issue, the simultaneous things that you can do are very controllable.
You can genuinely appreciate and recognise the efforts of your team members. This starts with saying, "Thank you."
You can look out for developmental opportunities (including projects) and offer them to your team members.
You can listen to their suggestions and genuinely take them on board and then get back to them about why their idea has/hasn't been implemented.
You can assign tasks to team members that truly reflect their talents while at the same time creating some 'stretch' for them. Of course, this means that you will have bothered to find out what their talents are!
You can create team celebrations to celebrate successes.
You can bother to remember the whole of life details that your team members have felt comfortable sharing with you (ie their partners and/or children's names, their birthday, special events in their life, their sporting teams and heroes etc.)
You can articulate how your team is contributing to your organisations vision and mission and help each person to 'see' how they are personally contributing to bringing these to life.
When times demand it you can make decisions that are timely and help the team to achieve its objectives.
These actions are all doable and are well within the control of a leader, irrespective of company culture.
It is my experience that when these activities are done with genuine intent, most people respond with an increase in self motivation and perform to a higher standard which is ultimately what leadership is trying to achieve.
How have you worked to increase the motivation of the members in your team?