I very much doubt the cable you dscribe is the right one. What you describe looks like this, right?
https://www.newegg.com/Product/Prod...m_re=fan_adapter_cable-_-12-423-172-_-Product
It has both a male and a female 4-pin Molex connector on it. You plug the male end into a female output from the PSU, and the female end of the adapter replaces the "used" output from the PSU, so it's a "pass-through" design. Then you plug the female connector from your fan into the adapter's male output. BUT you seem to think that you can connect the adapter's fan connector to a mobo fan header, and then plug the adapter's female Molex into the 4-pin male Molex on the little board to provide the board with power from the mobo. That can't work. The mobo header is male and the adapter's fan connector also is male, so they don't connect!
It appears that the little board is intended to provide power to several (up to 5) 3-pin fans from a single PSU female 4-pin Molex source. If you do it that way it will work, but all those fans will run at full speed all the time.
IF your wish is NOT to power fans that way, but to ensure that all your fans are powered AND speed-controlled by a mobo header, using the little board will require you to custom-modify a cable to get power from the mobo to the little board. But a better alternative would likely be to use a simple Splitter to connect the fans to the mobo header, and ignore that little board completely.
To do that you need to ensure you meet a particular limit. This applies no matter how you make the connection. A standard mobo fan header can supply up to 1.0 amps total to its combined fan load. You have three fans. Common fans used in desktop systems draw 0.1 to 0.25 amps each, so combining three using a Splitter into one mobo header is quite all right. BUT many higher-use systems (such as you may have) use higher-power fans with much higher amperage pull. That is why the rack system you have provides this little board to get power directly from the PSU that is not limited by the mobo header's limits. So you need to verify the amperage ratings for all three fans.
IF the total power consumption of those three fans is less than 1 amp, then you can use a simple Splitter. It does not matter whether you get one designed for 3-pin or 4-pin systems because both will work. Here are two example options.
https://www.newegg.com/Product/Prod..._re=coboc_fan_splitter-_-12-423-162-_-Product
https://www.newegg.com/Product/Prod..._re=coboc_fan_splitter-_-12-423-168-_-Product
The first is one with three outputs on a single cable, directly suitable for your system. The second is a unit with two outputs on separate 6" arms. One with 12" arms also is available. To use that, you need three of them. Then you plug two of them into the outputs of the third one, and the combined "stack" provides four outputs from one mobo header.
NOTE that each of these Splitters has one female connector to plug into a mobo header, and two or three male outputs. It has NO other connectors or arms. Do NOT buy a HUB. That is a different device, but unfortunately some are sold with the label "Splitter" and they are not. A HUB is different in that it also has an extra arm that must plug into a power output (either female 4-pin Molex or SATA Power Output) from the PSU. But a Hub can ONLY work with 4-pin systems, which you do not have.
If you do it this way, you MUST connect those 3-pin fans to either the CHA_FAN2 or CHA_FAN3 header. Those two headers use Voltage Control Mode, necessary for use with 3-pin fans. The CHA_FAN1 header uses PWM Mode and cannot control 3-pin fan speeds.
IF your check discloses that your fans use much higher currents so that they can't meet the limits of a mobo CHA_FAN header, post back here for alternatives.