It is a fact of life that divorce and even the idea of child support and court battles can bring a level of resentment into the relationship that is unparalleled and extremely destructive. If the adults are not mature enough to deal with it the battles and stresses the effects can spill over to the parenting role. The idea that you keep your children out of the battle is challenging, nearly impossible some days but you must do more than try to avoid slighting the other party in your parenting responsibility.
The child should not be involved in the role of parenting. They have no business knowing how much support is supposed to be paid, why the marriage broke up, why Mommy did X or why Daddy did Y. They must only understand that they are not the cause of the situation they are in. It's fair to explain to your children that their requests for clothes are best directed to the Custodial Parent (CP) as they are already funded for such expenses. It is also fair to explain that you don't have any more money for this toy or that, or even for a car because you have a commitment to pay support and that the support responsibility is the first priority, other items are luxuries. In the event the child questions your reasons for not being able to afford toy x or event z, your answer should be, "because I cannot afford it." If the child is confused about why you can't afford it when you work and their Mom (for example) can but doesn't work, you will be tempted to explain that your ex is bleeding you dry. This is not acceptable. I might suggest that you can explain where your money goes at a high level, including that you have a responsibility to pay support to ensure they have a safe and comfortable primary residence.
Watch what you say. The comments you make are potentially hurtful to all parties, and often a backlash by a child can result if they find out what you mean by an off-hand remark. Children today are smarter than you might think, they understand what you mean by "Children's Aid" or "Calling the Police" on their other parent. You cannot saddle them with this threat or information. The outcome can be unpredictable as they child may take it as an accusation of the parent it's directed at or a truth. Accusing the father of risking the child's life could lead the child to fear the father or gravitate to him, in his defense. It's best to keep such comments to yourself.
Dicipline is the responsibility of the natural parents. Shirking this responsibility through allowing the child to do whatever he/she wants or by passing on the bad-guy role to a new partner is unfair to the other parent and the child. The premise of my statement is that the custodial parent, having eliminated the negative role of dicipline, is unreasonably sainted by the child and the non-custodial parent is left with the role if diciplinary enforcement. If he fails to enforce either the resulting behavioural problems can produce a spoiled child who expects everything, with no understanding of responsibility. This is a poor beginning to life in today's society.
Your children are your first priority they deserve both parents in most cases and every effort should be made to support a loving relationship of each parent. Do not co-miserate with your child over what they perceive as unfair treatment, rather back up the parent while offering guidance on how to avoid the situation in the future. This does not mean support abuse, but don't involve the child in discussions about their Mom or Dad's parenting abilities. This extends to your new partner and family, the children should never be engraciated into the adult discussions of who did what to whom and why.