Whether covariates that are balanced at baseline should be put into propensity score matching I am performing a logistic regression under propensity score matching.
Before matching, the baseline characteristics of intervention group A and intervention group B are statistically described, where the covariates with clinical significance are “age, gender, whether or not have diabete, whether or not have hypertension, etc.”
We found that the covariates "age" and "diabete" are balanced before matching (p>0.05 and SMD >0.1). But I am sure both they are clinically significant confounders to outcome with references evidence.
So I am not sure whether I should put in such covariates that are balanced at baseline when calculating the propensity score? Or should covariates be taken into account in the PSM regardless of whether they are balanced (without satistical difference) at baseline or not?
Confusingly, if these variables that are balanced at baseline are not put into PSM, there is a very high probability that they will not be balanced any more after matching (p<0.05 or SMD>0.1).

The answer to this is the same as it is to most questions about how to balance covariates: try both and choose the better one. General principles may help guide these decisions, but none of them apply exactly to your dataset, which is why you should try many options and select the best one. You claim one of your specifications yielded better balance; choose that one (as long as the effective sample size is not too degraded).
There is no law about the relationship between pre-matched balance and post-matched balance. Anything can happen, and it depends on the unique features of your dataset. It's common for well-balanced covariates to remain well-balanced whether they are included in the balancing or not, but if you can guarantee balance using some method then do so.
Also remember that bias in the effect estimate is a function of imbalance on prognostic variables. This is the main motivation for balancing. If you notice imbalance in a (pretreatment) prognostic variable, seek to remove that imbalance, however you can do that (without involving the outcome).
