As team manager, if one of your team members' performance is being affected by something the company is doing, then that is your problem. This is true whether it is salary, work environment, or not giving the proper tools. You should follow this up with the company.
Also, your team members' motivation and morale are absolutely your problem. An unmotivated team member will perform poorly, and complaining about being 'unprofessional' won't change that. In my management style if your team is suffering, it's your job to do whatever you can to alleviate it. If you don't care about them you are not doing your job properly. You may not be able to do anything, but you should at least try.
Start with the obvious. Not paying someone on time is illegal. You should absolutely talk to HR, or other management, about this. Cutting salary is also a problem. In any country except the US, doing this without the employee's agreement would be illegal. It may be the missed payments are glitches due to the salary change, but you should at least find out the truth.
You should respond by going to management and telling them what you know - that you have an underperforming team member, who is impacting the team's ability to deliver, and that the reasons are what you were told. You should find out why the salary was cut, why there are late payments, and why they are not doing the job they were hired for. That's not an unreasonable request. If a member of your team was not performing to expectations, then you should have been included in the process.
If the reason is that management is trying to get rid of the employee, then you have two choices. Either play the loyal company follower and say nothing, or quietly and unofficially pass this on to the employee and encourage them to find work elsewhere.
If the reason is that the employee was underperforming, and salary has been reduced to reflect that, then you need to tell the employee that this is the situation, and they need to accept the new role and step up to do it well. The alternative should be implied.