I had the idea that it could be that another spike lies in the range of plus/minus 12 of your spike-position. in this case you should use the following

    spikepos = {5, 8, 40, 70, 95};
    xx = Range[100];
    yy = Table[
       If[Or @@ Thread[spikepos == j], 50, RandomReal[{-1, 1}]], {j, 1, 100}];
    bigmat = Transpose[{xx, yy}];
    ListLinePlot[bigmat, PlotRange -> All]
    (*now construct smallmat*)
    smallmat = bigmat[[#]] & /@ spikepos

    (*with smallmat given,find the positions of peaks in bigmat*)
    pos = Flatten[Position[bigmat, #] & /@ smallmat]
    (*now get the datapoints around these positions*)
    n = 12;(*this should be replaced by 12 as you wanted*)
    pos1 =  Table[j, {j, # - n, # + n}] & /@ pos;
    % // MatrixForm
    pos2 = Function[x, Select[x, And[# > 0, # <= Length[bigmat]] &]] /@  pos1;
    pos2 // MatrixForm
    pos3 = Complement[#, pos] & /@ pos2;
    sublists = bigmat[[#]] & /@ pos3
    flists = Interpolation[#] & /@ sublists
    xvals = bigmat[[#, 1]] & /@ pos
    replacements = Table[{xvals[[i]], flists[[i]][xvals[[i]]]}, {i, 1, Length[pos]}]
    newbigmat =  ReplacePart[bigmat, Transpose[{pos, replacements}] /. {x_, y_} -> x -> y];
    ListLinePlot[newbigmat, PlotRange -> All]

OK.

I assume your matrices are x,y values. May be you don't want to just throw away your spikes, but substitute them by appropriate means of surrounding values (here 2 to the left and 2 to the right, at least far from the edges). Look at this

bigmat = Table[{x[i], y[i]}, {i, 10}]
smallmat = {{2, 2}, {5, 5}, {9, 9}} /. {a_, b_} -> {x[a], y[b]}
(*positions of spikes *)
index = Function[{x}, Position[bigmat, x][[1, 1]]] /@ smallmat
(*values around these positions*)
sublists = DeleteCases[
    bigmat[[Max[1, # - 2] ;; Min[Length[bigmat], # + 2]]],
    {x[#], y[#]}] & /@ index
(* calculate the means*)
meanvals = 
 Function[{x}, Total[Transpose[x][[2]]]/Length[x]] /@ sublists
(* add the x-vals*)
corrvals = Transpose[{x[#] & /@ index, meanvals}]
rule = Table[smallmat[[i]] -> corrvals[[i]], {i, 1, Length[smallmat]}]
bigmatnew = bigmat /. rule

about the average method you wrote, I think it's better to not consider the value of point which we want to replace in the calculation of average so I modified it to just consider the points around the specific point we want to replace. again thank you for your help.

bigmat = Table[{x[i], y[i]}, {i, 10}];
smallmat = {{2, 2}, {5, 5}, {9, 9}} /. {a_, b_} -> {x[a], y[b]};
(*positions of spikes*)
index = Function[{x}, Position[bigmat, x][[1, 1]]] /@ smallmat;    
(*values around these positions*)
    sublists1=DeleteCases[bigmat[[Max[1,#-2];;Min[Length[bigmat],#-1]]],{x[#],y[#]}]&/@index ;
    sublists2=DeleteCases[bigmat[[Max[1,#+1];;Min[Length[bigmat],#+2]]],{x[#],y[#]}]&/@index ;
    (*calculate the means*)
    meanvals1=Function[{x},Total[Transpose[x][[2]]]/Length[x]]/@sublists1 ;
    meanvals2=Function[{x},Total[Transpose[x][[2]]]/Length[x]]/@sublists2 ;
    (*add the x-vals*)
    corrvals=Transpose[{bigmat[[#]][[1]]& /@index,(meanvals1+meanvals2)/2}] ;
    rule=Table[smallmat[[i]]->corrvals[[i]],{i,1,Length[smallmat]}] ;
    bigmatnew=bigmat/.rule 

A bit more instructive would be to use

xx = Range[10];
yy = Table[
   If[Or @@ Thread[{2, 5, 9} == j], 300, 
    j^2 + RandomReal[{-10, 10}]], {j, 1, 10}];
bigmat = Transpose[{xx, yy}]

Thank you.

It occurred to me that it is perhaps better to use a linear interpolation between two points neighbouring the spikes instead to take the mean of several points around the spike? What do you think?

Have a look at the following code

bigmat = Table[{x[i], y[i]}, {i, 10}]
smallmat = {{2, 2}, {5, 5}, {9, 9}} /. {a_, b_} -> {x[a], y[b]}
(*positions of spikes *)
index = Function[{x}, Position[bigmat, x][[1, 1]]] /@ smallmat
index1 = {Max[1, # - 1], #, Min[Length[bigmat], # + 1]} & /@ index
(*values around these positions*)
val = Map[bigmat[[#]] &, index1, {2}]
(* do the interpolation *)
nval = Function[{a, b, c},
   {b[[1]],  a[[2]] (b[[1]] - c[[1]])/(a[[1]] - c[[1]]) + c[[2]] (b[[1]] - a[[1]])/(c[[1]] - a[[1]])}] @@@ val
rule = Table[smallmat[[i]] -> nval[[i]], {i, 1, Length[smallmat]}]
bigmatnew = bigmat /. rule
% // Simplify

You should test it and decide which method is more appropriate.

And in fact I don't understand your issue with Complement:

Length[Complement[bigmat, smallmat]]
Length[bigmat]
Length[smallmat]

Are there duplicates in your data?

Hi sir, Thank You very much for your response, yes I think the Function I need is the function you said, Complement. I had a problem when I used this function that is I checked and the intersection of two matrices size is equal to smaller matrix means that every object in smaller one is an object of bigger one, but this Complement function gives me 1812 result points instead of the difference of this two matrices namely 2086-169 = 1917 points; is there any problem I didn't get it or no?? Again thank you for your help.