稍微修改一下你的代码，它应该可以工作。你不能以那种方式使用strcat，因为strcat的第二个参数接受const char *，而你有char，所以你的代码不会编译。
这里还值得注意的是：我假设函数add将为fnme和anme创建一个deepcopy，否则，您不能在这里简单地释放它们。

while ((character = fgetc(fp)) != EOF) {
    char *fnme, *anme;
    fnme = malloc(100 * sizeof(char));
    anme = malloc(100 * sizeof(char));

    int i = 0;
    while (character != ' ') {
        fnme[i++] = character;
        character = fgetc(fp);
    }
    fnme[++i] = '\0';   // NULL-terminate

    i = 0;
    while (character != '\n') {
        anme[i++] = character;
        character = fgetc(fp);
    }
    anme[++i] = '\0';  // NULL-terminate

    add(head, fnme, anme);   // Assume 'add' will make deep copy of both fname and 
                             // anme, otherwise you cannot free them here.
    free(fnme);
    free(anme);

}

strcat绝对不适合使用。正如编译器应该告诉你的那样，你首先传入了错误类型的变量。它需要2个字符串，而character只是一个char。
C中的字符串是以NUL结尾的，并且刚刚为anme和fnme分配了内存，它们都不会被初始化，所以不会包含NUL结尾，所以你会遇到未定义的行为。
相反，只要把它们当作数组，在读入字符时存储它们，然后记住在阅读所有字符后以NUL终止它。

int count = 0;

do{  
    fnme[count++] = character;
    character = fgetc(fp);
}while(character != ' ');
fnme[count]='\0'; // Need to NUL terminate the string

count = 0; // Remember to reset count to 0
do{  
    anme[count++] = character;
    character = fgetc(fp);
}while(character != '\n');
anme[count]='\0'; // Need to NUL terminate the string

此方法还允许您检查count是否超出了分配的大小，在本例中为99，因为NUL字符需要最后一个空间。

你在add()函数中有一些错误。这是完整的解决方案。顺便说一句，由于这是一个后续问题，你最好在新的帖子中问它，而不是更新原来的帖子。

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

typedef struct node{
  char *forame;
  char *aftername;
  struct node *next;
}node;

void add(node **head, char* fnme, char* lnme){
  node *new_node;
  new_node = (node*)malloc(sizeof(node));
  new_node->forame = (char*)malloc(100*sizeof(char));
  new_node->aftername = (char*)malloc(100*sizeof(char));

  strcpy(new_node->forame, fnme);
  strcpy(new_node->aftername, lnme);

  if (*head == NULL){
    *head = new_node;
    new_node->next = NULL;
    return;
  }

  node *current;
  current = *head;

  while(current->next != NULL){
     current = current->next;
  }

  current->next = new_node;
  new_node->next = NULL;
}

void print(node *head){
   node *current;
   current = head;

  while(current != NULL){
      printf("%s %s\n", current->forame, current->aftername);
      current = current->next;
  }
}

int main() {

  node *head = NULL;
  char character;

  FILE *fp;
  fp = fopen("input.txt", "r");

  while ((character = fgetc(fp)) != EOF) {
      char *fnme, *lnme;
      fnme = (char*)malloc(100 * sizeof(char));
      lnme = (char*)malloc(100 * sizeof(char));

      int i = 0;
      while (character != ' ') {
          fnme[i++] = character;
          character = fgetc(fp);
      }
      fnme[++i] = '\0';

      i = 0;
      while (character != '\n') {
          lnme[i++] = character;
          character = fgetc(fp);
      }
      lnme[++i] = '\0';

      add(&head, fnme, lnme);

     free(fnme);
     free(lnme);
  }

  print(head);
  return 0;
}