declare @ TimeVal time,
           @ TotalTime time; -- spaces between @ and rest of variable names to allow posting of code
set @TimeVal = '08:05:44.0000000';
set @TotalTime = '00:13:00.0000000'
select @TimeVal, @TotalTime, dateadd(ss, ((DATEPART(HOUR,@TotalTime) * 60) + DATEPART(MINUTE,@TotalTime) * 60) + DATEPART(SECOND,@TotalTime), @TimeVal);

-- note how this starts off by creating a table structure and putting data into it?
-- In order to get people willing to help, this is a prerequisite on your part.
-- see the link in my signature for how to do this.
declare @VerizonDetails table ([Connect Time2] TIME, [totalTime2] TIME)
insert into @VerizonDetails
values ('08:05:44.0000000', '00:13:00.0000000'),
       ('08:05:57.0000000', '00:01:00.0000000'),
       ('09:07:42.0000000', '00:03:00.0000000'),
       ('09:07:46.0000000', '00:09:00.0000000'),
       ('09:08:08.0000000', '00:01:00.0000000')

-- add the two times together, first converting them to datetimes
select *, convert(time, CONVERT(datetime, [Connect Time2]) + CONVERT(datetime, totalTime2))
  from @VerizonDetails

declare @ TimeVal time,
        @ TotalTime time;  -- space added between @ and variable name to allow code to post
set @TimeVal = '08:05:44.0000000';
set @TotalTime = '00:13:00.0000000'
select  @TimeVal,
        @TotalTime,
        dateadd(ss, ((DATEPART(HOUR,@TotalTime) * 60) + DATEPART(MINUTE,@TotalTime) * 60) + DATEPART(SECOND,@TotalTime), @TimeVal),
        @TimeVal + @TotalTime;

Msg 8117, Level 16, State 1, Line 6
Operand data type time is invalid for add operator.

declare @ TimeVal time,
        @ TotalTime time;  -- space added between @ and variable name to allow code to post
set @TimeVal = '08:05:44.000';
set @TotalTime = '00:13:00.000';
select  @TimeVal,
        @TotalTime,
        dateadd(ss, ((DATEPART(HOUR,@TotalTime) * 60) + DATEPART(MINUTE,@TotalTime) * 60) + DATEPART(SECOND,@TotalTime), @TimeVal),
        convert(time, (convert(datetime, @TimeVal, 108) + convert(datetime, @TotalTime, 108)), 108);