Pau Gasol might not always give it his all on the court (at least according to D'Antoni) -- but he definitely goes the extra mile when he visits sick kids ... TMZ Sports has learned.

The Lakers star went to Children's Hospital of LA yesterday -- and we're told he spent over 4 hours hanging out with kids ... signing personalized t-shirts and basketballs he brought for them.

A rep for the hospital says the kids and their parents were blown away by Pau's bonding skills ... and that he makes the trip at least three times a year.

Pau is reportedly off the trading block for the Lakers -- despite his most recent public spat with Mike D'Antoni ... which is good news for the kids at CHLA.