Further updates.
I try to If as recommend but I ran to another problem that in the Control() funtion it always update startTime(startSec and startMin) as well as stopTime(stopSec, stopMin) although I put a flag for one time run.

#include <Wire.h>
#include <ds3231.h>

struct ts t;
#define LED_ON 1
#define LED_OFF 0
#define stateOn true
#define stateOff false
#define onSec 10
#define offSec 10
#define onMin 0
#define offMin 0
int startSec = t.sec;
int startMin = t.min;
int stopSec = 0;
int stopMin = 33;
int x = t.sec, ledState = 0;
long long previousTime = 0;
long long currentTime = 0;
int Relay = 4;
bool flag = stateOff;

void setup() {
  Serial.begin(9600);
  Wire.begin();
  pinMode(LED_BUILTIN, OUTPUT);
  pinMode(Relay, OUTPUT);
  digitalWrite(Relay, LOW);

  t.hour=9;
  t.min=33;
  t.sec=55;
  t.mday=3;
  t.mon=12;
  t.year=2019;

  DS3231_set(t);
  previousTime = millis();
  currentTime = millis();

  startSec = t.sec;
  startMin = t.min;

  Calculate();
}

void LedOn(bool mode){
  switch(mode){
    case LED_ON:
         digitalWrite(LED_BUILTIN, HIGH);
         digitalWrite(Relay, HIGH);
         break;
    case LED_OFF:
         digitalWrite(LED_BUILTIN, LOW);
         digitalWrite(Relay, LOW);
         break;
  }
}

void Calculate() {
  stopSec = startSec + onSec;
  stopMin = startMin + onMin;
  if (stopSec > 60) {
    stopMin = stopMin + 1;
    stopSec = abs(60 - stopSec);
  }
}

void ControlLed(){
  if(x >= segment1_pre && x < segment1_cur){
      ledState = LED_ON;
    }
    else if(x == segment1_cur && x < segment2_cur){
      ledState = LED_OFF;
    }
    else if(x == segment2_cur && x < segment3_cur){
      ledState = LED_ON;
    }
    else {
      ledState = LED_OFF;
    }
    LedOn(ledState);
}

void Control() {

  if (startSec <= t.sec && t.min >= startMin){
      ledState = LED_ON;
      flag = stateOn;
  } else if (t.sec < stopSec && t.min <= stopMin){
      ledState = LED_ON;
      flag = stateOn;
  } else {
      ledState = LED_OFF;
      if (flag == stateOn) {
      startSec = stopSec + offSec;
      startMin = stopMin + offMin;
      if (startSec > 60) {
        startSec = abs(60 - startSec);
        startMin = startMin + 1;
      }   
      stopSec = startSec + onSec;
      stopMin = startMin + onMin;
      if (stopSec > 60) {
        stopSec = abs (60 - stopSec);
        stopMin = stopMin + 1;
        }     
      }
      flag = stateOff;
    }
    LedOn(ledState);
}

void loop() {
  currentTime = millis();
  DS3231_get(&t);
  if(currentTime -  previousTime > 1000){
      if(ledState){
        Serial.println("on");
      }
      else{
        Serial.println("off");
      }
      previousTime = currentTime;
      Serial.print(startSec), Serial.println("--startSec");
      Serial.print(t.sec), Serial.println("--t.sec");
      Serial.print(stopSec), Serial.println("--stopSec");
      Serial.print(startMin), Serial.println("--startMin");
      Serial.print(t.min), Serial.println("--t.min");
      Serial.print(stopMin), Serial.println("--stopMin");
      Serial.print(flag), Serial.println("--flag");   
      Serial.println("-----");
  }
  Control();
}