Holding buttons (keyboard)

The root problem is that you send each several ms new key event and on receiver each time you get this event, you call Keyboard.press and right away Keyboard.release. So, it behaves as several key press-releases.

At sender side you should send 2 types of events:

1. Key pressed (and which button)
2. Key released (and which button)

At receiver side you should

1. When you receive "press" event, call Keyboard.press, and don't call Keyboard.release
2. When you receive "release" event, call Keyboard.release

Edit: I've made changes to your source code, please, try it

Transfer

#include <RF24Network.h>
#include <RF24.h>
#include <SPI.h>

RF24 radio(8,7);
RF24Network network(radio);

const uint16_t this_node = 03;
const uint16_t other_node = 00;

unsigned int laser_state=0;
const unsigned long interval = 5;
unsigned long last_sent;

unsigned int b1=1;
unsigned int b2=1;
unsigned int b3=1;
unsigned int b4=1;

unsigned int d1=10;
unsigned int d2=11;
unsigned int d3=12;
unsigned int d4=13;

//TR1
int btn1 = 3;
int btn2 = 4;
int btn3 = 5;
int btn4 = 6;

void setup(void)
{
//  Serial.begin(9600);
  pinMode(btn1,INPUT_PULLUP);
  pinMode(btn2,INPUT_PULLUP);
  pinMode(btn3,INPUT_PULLUP);
  pinMode(btn4,INPUT_PULLUP);

  SPI.begin();
  radio.begin();
  network.begin(90,this_node);
}


void loop(void){
  network.update();

  unsigned long now = millis();

  if (now - last_sent >= interval)
  {
    last_sent = now;

    auto newB1 = digitalRead(btn1);
    auto newB2 = digitalRead(btn2);
    auto newB3 = digitalRead(btn3);
    auto newB4 = digitalRead(btn4);

    unsigned int packet[2];

    if(newB1 != b1){
//      Serial.println("F1");
      packet[0] = d1;
      packet[1] = newB1;
      RF24NetworkHeader header(other_node);
      bool ok = network.write(header,packet,sizeof(packet));
    }

    if(newB2 != b2){
//      Serial.println("F2");
      packet[0] = d2;
      packet[1] = newB2;
      RF24NetworkHeader header(other_node);
      bool ok = network.write(header,packet,sizeof(packet));
    }

     if(newB3 != b3){
//      Serial.println("Ctrl");
      packet[0] = d3;
      packet[1] = newB3;
      RF24NetworkHeader header(other_node);
      bool ok = network.write(header,packet,sizeof(packet));
    }

    if(newB4 != b4){
//      Serial.println("Shift");
      packet[0] = d4;
      packet[1] = newB4;
      RF24NetworkHeader header(other_node);
      bool ok = network.write(header,packet,sizeof(packet));
    }

    b1 = newB1;
    b2 = newB2;
    b3 = newB3;
    b4 = newB4;
  }
}

Receiver

#include <Keyboard.h>
#include <RF24Network.h>
#include <RF24.h>
#include <SPI.h>

RF24 radio(8,7); // (ce,cs)
RF24Network network(radio);

const uint16_t this_node = 00;
const uint16_t other_node = 03;

unsigned int laser_state=0;
const unsigned long my_interval = 500;
unsigned long last_sent;


void setup(void)
{
  Serial.begin(115200);
  pinMode(9,INPUT_PULLUP);

  SPI.begin();
  radio.begin();
  network.begin(90,this_node);

  pinMode(4,OUTPUT);
  digitalWrite(4,LOW);
}


void loop(void){
  digitalWrite(4,LOW);
  network.update();

  unsigned int income[2] = {0, 0};
  
  if ( network.available() ) {
    RF24NetworkHeader header;
    network.read(header,&income,sizeof(income));

    auto keyCode = income[0];
    auto keyPress = !income[1];

    if(keyCode == 10){
      Serial.println("f1");
      
      if (keyPress) {
        Keyboard.press(KEY_F1);
      } else {
        Keyboard.release(KEY_F1);
      }
    }

    if(keyCode == 11){
      Serial.println("f2");
      
      if (keyPress) {
        Keyboard.press(KEY_F2);
      } else {
        Keyboard.release(KEY_F2);
      }
    }

    if(keyCode == 12){
      Serial.println("ctrl");

      if (keyPress) {
        Keyboard.press(KEY_LEFT_CTRL);
      } else {
        Keyboard.release(KEY_LEFT_CTRL);
      }
    }

    if(keyCode == 13){
      Serial.println("shift");

      if (keyPress) {
        Keyboard.press(KEY_LEFT_SHIFT);
      } else {
        Keyboard.release(KEY_LEFT_SHIFT);
      }
    }

    keyCode = 0;
  }

  unsigned long now = millis();

  if ( now - last_sent >= my_interval  )
  {
//    Serial.print("pin ");
//    Serial.println(digitalRead(9));
    last_sent = now;
    laser_state = digitalRead(9);

    RF24NetworkHeader header(other_node);
    bool ok = network.write(header,&laser_state,sizeof(laser_state));
  }
}